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Questions tagged [indeterminate-forms]

If the expression obtained after any substitution during limit analysis does not give enough information to determine the original limit, it is known as an indeterminate form.

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Solving a combined limit with an $1^{\infty}$ form nested inside a 0×∞ form

I came across this limit problem: $\lim _{x \rightarrow \infty}\left\{\left(\frac{x+1}{x-1}\right)^x-e^2\right\} \cdot x^2$ Plugging this into desmos, one can see that the limit approaches $\frac{2 e^...
Afsheen's user avatar
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Can someone explain when a limit is defined?

Why is it that the following limit is defined if $\lim_{x\to a}f(x) = \lim_{x\to a}g(x) = 0$? $$\lim_{x\to a}\frac{f(x)}{g(x)}$$ In contrast, the limit isn't defined if $\lim_{x\to a}f(x) \neq 0$ but $...
Aryaan's user avatar
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Can we say that $1^\infty$ is indeterminate because we can't classify $\infty$ as even or odd?

While studying indeterminate forms I like many others had used this post to understand why $1^\infty$ is indeterminate. Today I thought of a new and perhaps simpler argument to explain why this is so ...
Madly_Maths's user avatar
5 votes
3 answers
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Evaluate $\lim\limits_{x\to\infty}x\!\left[2x\!-\!\left(x^3\!+\!x^2\!+\!x\right)^{\!\frac13}\!\!-\!\left(x^3\!-\!x^2\!+\!x\right)^{\!\frac13}\right]$ [closed]

Evaluate $\lim\limits_{x\to \infty}x\left[2x-\left(x^3+x^2+x\right)^{\frac{1}{3}}-\left(x^3-x^2+x\right)^{\frac{1}{3}}\right]$ My Approach: Formula I used $(1+x)^{n}=1+nx+\frac{n(n-1)}{2!}x^2+.....\...
mathophile's user avatar
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2 votes
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How to do this limit $L=\lim_{(x,y) \to (1,4)} \frac{y^2 - 4xy}{y^2 - 16x^2}$

Evaluate : $$L=\lim_{(x,y) \to (1,4)} \frac{y^2 - 4xy}{y^2 - 16x^2}$$ My Work : We can cancel $(y-4x)$ from the numerator and denominator provided $y \neq 4x$ and this comes out as $1/2$. When the ...
Thomas Merrells's user avatar
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If the limit $\lim\limits_{x\to 0}{\frac{\sin 3x}{x^3} + \frac{a}{x^2}+b}$ exists and equals $0$ then what can $a$ and $b$ be?

Let $$L=\lim\limits_{x\to 0}{\frac{\sin 3x}{x^3} + \frac{a}{x^2}+b}=0$$ given that $a,b \in \mathbb R$ and are finite. I tried the following approach, We know, $\lim\limits_{x\to 0}{\frac{\sin 3x}{3x}}...
Jesko's user avatar
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How to compute the following limit? $\lim\limits_{x\to\ 0^+} {\frac{x-\lfloor x \rfloor}{x+\lfloor x \rfloor}}$ [closed]

$$\lim\limits_{x\to\ 0^+} {\frac{x-\lfloor x \rfloor}{x+\lfloor x \rfloor}}$$ Here, $\lfloor x \rfloor$ represents the floor of $x$. I tried using a graphing calculator (desmos) to plot the function $...
Jesko's user avatar
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Limit evaluation, de l'Hopital seems not working

I have to evaluate \begin{equation} L=\lim_{x\to 0} \frac{1-e^{\frac{x^2}{2}}\cos x}{2\sin^2x -x \arctan2x} \end{equation} I tried with de l'Hopital's rule but it seems not working, am I missing ...
MStocchi's user avatar
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Why is the graph of function behaving in a seemingly erratic fashion?

I was recently trying to plot the graph of the function $f(x)$. This function is defined as follows: $ f(x) = \frac{\sin{3x} - 3\sin{x}}{(\pi - x)^3} $ I first plotted the numerator ($A(x) = \sin{3x} -...
rohan843's user avatar
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How to get the limit of the sequence $(\sqrt{n^2+n}-\sqrt{n^2-1})$? [duplicate]

My Attempt $$ \sqrt{n^2+n} - \sqrt{n^2-1} = \sqrt{n+1} \, \bigl(\sqrt{n}-\sqrt{n-1} \bigr) $$ Then I tried to apply the sandwich theorem in some way but failed. Important Note Please do not solve the ...
IncredibleSimon's user avatar
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Using L'Hospital's Rule for Fourier's series

The origin of this question comes from watching this YouTube video at 8:00. The equation to evaluate is $$f(x)=\frac{1}{i(j-k)}e^{i(j-k)x}|_{-\pi}^{\pi}$$ However, this equation can be evaluated ...
user97662's user avatar
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Does $ \lim_{x \to 0^+} (1 + x)^{\ln x}$ exist? [closed]

Does $\displaystyle \lim_{x \to 0^+} (1 + x)^{\ln x}$ exist? I do not have any attempts solving it because I do not know how to transform it into the form in which I use l'Hôpital's rule.
wika27's user avatar
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Can the following $\infty - \infty$ limit be solved without using l'Hôpital rule?

I know that some of the $\infty - \infty$ limits can be solved without using l'Hopital's rule. In such cases, you would usually be able to either rationalize (or derationalize if that is what its ...
Spime's user avatar
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3 answers
108 views

Bypassing the indeterminate form $0\cdot \infty$

Trying to calculate out the limit \begin{array}{rcl} \lim_{x \to +\infty } (e^{x^{2}\sin \frac{1}{x}}-e^{x})&& \\ \end{array} I come up with the indeterminate form $0\cdot \infty$ as ...
Κωνσταντίνος Παναγιώτου's user avatar
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1 answer
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How can we multiply&divide the expression by its conjugate in the $\infty-\infty$ indeterminate case? Is it not $\frac{\infty}{\infty}$?

Question $$\lim _{x \rightarrow \infty} \left(x-\sqrt{x^2+5 x}\right)$$ To evaluate the limit, we multiply and divide the expression by its conjugate. First Question But since $x \rightarrow \infty$, ...
1_student's user avatar
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3 answers
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$\infty-\infty$ indeterminate case

The methods yield two different answers. Could you explain the reason clearly and in detailed? Question: $$\lim_{{x \to \infty}} \left( \sqrt{x^2 + 6x + 14} - (x+1) \right) = ?$$ Solution: Method 1: ...
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Indeterminate forms other than the 7 common ones

The following 7 indeterminate forms are all I can find in any calculus books: $$\frac{0}{0}, \frac{\infty}{\infty}, 0 \cdot \infty, \infty - \infty, 0^0, \infty^0, 1^\infty.$$ For example, by $\frac{0}...
Joseph's user avatar
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For smooth $f$ on $[0,1]$ such that $f^{(k)}(0)=0$ for all $k\in\mathbb{N}$, is it true that $\lim_{x\to0}\frac{xf'(x)}{f(x)}=\infty$?

Got asked this question and I got a bit surprised at how messy it became. Suppose $f$ is a smooth function on $[0,1]$ such that the $k^{\text{th}}$ derivative $f^{(k)}(0)=0,\forall k\in \mathbb{N}$. ...
Ace's user avatar
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Limit of a sequence: $\displaystyle \lim_{n\to\infty}\sqrt{n}\left[\frac{e^{\sqrt{n+1}}}{e^{\sqrt{n-1}}}-1\right]?$ [closed]

The sequence in question: $$(u_{n})_{n\ge 0}=\sqrt{n}\left[\dfrac{e^{\sqrt{n+1}}}{e^{\sqrt{n-1}}}-1\right]$$ I see that there is an indeterminate form that must be lifted to calculate the limit, but I'...
Looky1173's user avatar
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For which values ​of a this limit is zero?

I need to find the values of $a$ for which the following limit is $0$ : $$L = \lim _{x\rightarrow 0^{+}}\frac{\sin(\log( 1+3x)) -e^{3x} +\cos x}{(\sin x)^{3\alpha }}$$ It is an indeterminate form of ...
W. White's user avatar
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Proof of $\lim_{n\to \infty}(1 + \frac{b}{n})^n = e^b$ [duplicate]

I found one definition of $e$ to be $\lim_{n\to \infty}(1 + \frac{1}{n})^n = e$ and I checked it with a graphing calculator to be true. However the $1$ in the numerator is a special case, where it ...
Gustamons's user avatar
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4 answers
105 views

How do we evaluate $\lim_{x\to\infty} \left(\frac{x-c}{x+c}\right)^x$?

$\lim_{x\to\infty} (\frac{x-c}{x+c})^x=\frac{1}{4}$ My teacher did the following steps: $\lim_{x\to\infty} \ln{(\frac{x-c}{x+c})^x}=\ln{\frac{1}{4}}$ $\lim_{x\to\infty} x\ln{\frac{x-c}{x+c}}=\ln{\frac{...
Kai Lang's user avatar
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3 answers
58 views

Convergent values of series containing variables approache zero

Example 7 in Thomas' Calculus Early Transcendentals 14th edition p.614 demonstrates how to use Taylor series to find a limit involving an indeterminate form: $$ \lim_{x\to 0}\left(\frac{1}{\sin(x)} - \...
Tran Khanh's user avatar
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1 answer
441 views

Electric Field Due to uniformly charged infinitely long wire

Calculate the field at a point P at distance r from infinitely long wire with charge density $\lambda$ Generally the electric field in this case at point P, distance r from the wire is derived using :...
Aurelius's user avatar
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4 votes
3 answers
530 views

L'Hospital's Rule of infinity over infinity

I am troubled for understanding the L'Hospital's Rule of $\infty/\infty$ : $$\lim_{x\rightarrow a}\frac{f(x)}{g(x)}= \lim_{x\rightarrow a}\frac{f^\prime(x)}{g^\prime(x)} \tag{1}$$ where $\lim_{x\...
Daren's user avatar
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4 answers
278 views

Difference between 0+0+0+0...infinite times and 0 multiplied by infinity. [closed]

The measure/size of set I came across that the measure of a set that is formed by intersection of countably infinite disjoint sets of measure 0 each, will be the sum of the measures of all the sets = ...
theDemid's user avatar
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42 views

On the Divergence of an Improper Integral where the Integrand becomes unbounded in the Neighbourhood of a Point inside the Interval of Integration

I have been having some trouble with the way to determine the convergence/divergence of an improper integral such as this: $$\int_0^a \frac{\arctan{(x)}}{1-x^3}dx,$$ where $a>1.$ It is evident that ...
Barbatulka's user avatar
1 vote
3 answers
61 views

Tricky limit on the open first quadrant: $\lim_{(x,y) \rightarrow (0,0)} -x\log{y} $.

Consider $ f:(0,1) \times (0,1) \rightarrow \mathbb{R} $ where$$ f(x,y)=-x\log{y} $$ I am trying to prove whether $$ \lim_{(x,y) \rightarrow (0,0)}{f(x,y)}=0$$ My current idea is as follows: Let $ p:(...
Matt Szuromi's user avatar
1 vote
4 answers
72 views

Limits of indeterminate forms $(-\infty)(0)$

Can anybody give me a detailed solution on how $$\lim_{x \to \infty} -x\left(1-e^{-\frac{1}{1+x}}\right) = -1?$$ I understand that separately the limit of each factor is $(-\infty)(0)$ but I could not ...
user1178472's user avatar
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0 answers
40 views

Limit of a indeterminate function is always can be calculated?

In class, a teacher made the following statement: Whenever the limit of a real function of a real variable gives $\frac{0}{0}$ for $x \to a$, the limit can be calculated (via L'Hopital for example). I ...
Thiago Alexandre's user avatar
4 votes
2 answers
222 views

Advanced methods to explain indeterminate forms

I know this post may be covering a subject that is considered 'low quality' but I wanted to try and cover it in a more advanced manner (before writing I also searched if there were duplicate posts). I ...
Math Attack's user avatar
1 vote
4 answers
266 views

Why $ 0^{\infty} $ isn't indeterminate form?

I was trying to calculate $$ \lim _{x \rightarrow 0} x^{\frac{1}{x}} $$ I know left hand limit is not equal to right hand limit, hence limit doesn't exist. But I was trying to get their values as well....
Sohit Jatain's user avatar
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2 answers
223 views

Find the $\lim_{x\to 1} x^{\frac{1}{x^2-1}}$ without applying L'Hôpital's rule

Professor wants me to find the limit $$ \lim_{x\to 1} x^{\frac{1}{x^2-1}} $$ without using L'Hôpital and even gave the following advice: Rewrite the exponent as a product of sum and difference, make ...
Mateus's user avatar
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2 votes
1 answer
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Find Limit : $\lim_{x\to-∞}\frac {1}{xe^{x}}$

If I'm doing everything correctly, I get to 1/(-∞×0) where -∞×0 is undefined, so I don't know what to do. Also, I cannot do a series expansion because x is approaching infinity. Can someone please ...
Mihailo Mitrović's user avatar
1 vote
1 answer
95 views

Can indeterminate form be used as a critical point?

If we have a derivative, such as $(e^x-1)/x$, could we test the value of $x$ at $0$ to see it it’s a relative minimum or maximum (using the first derivative test) at that point? In other words, is it ...
Zach A.'s user avatar
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2 answers
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L'Hospital rule for $\frac{0}{0}\cdot\frac{a}{0}$ [closed]

Is there a solution to use the rule of L'Hospital for a boundary value problem like $$\lim_{x\to1}\left(\dfrac{x^{2}-1}{x-1}\cdot\dfrac{a}{x^{3}-1}\right)$$ where $a> 0$ ? I know how to solve $\...
maxwell1902's user avatar
1 vote
1 answer
68 views

How do you solve $\lim_{n\to\infty}\left(\frac{\log(n+1)}{\log(n)}\right)^{n\log(n)}$ [closed]

$\lim\limits_{n\to\infty}\left(\dfrac{\log(n+1)}{\log(n)}\right)^{n\log(n)}$ As it’s an indetermination, I’ve tried to do $\exp\left(\lim\limits_{n\to\infty}(n\log(n))\cdot(\log(n+1)/\log(n)-1)\right)$...
Zoe's user avatar
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2 votes
1 answer
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About limits of a particular form

I was talking with some colleagues about why $0^0$ cannot be simply just defined as $1$. Topology came out, set theory came out but most of all a friend of mine said this: "there is a theorem ...
Heidegger's user avatar
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Limits using L'Hopital's Rule

I tutor for AP Calc AB, it's been a couple years since I last took calculus though. My student had a question with a limit using L'Hopital's Rule, with $\infty/\infty$. The question was: $$\lim _{x \...
calvinl40's user avatar
-2 votes
1 answer
29 views

Showing that this piecewise defined function is infinitely differentiable [duplicate]

Let $a<b$ and $f(x) = 0$ if $x$ is outside $(a,b)$ and $f(x) =e^{-\frac{1}{x-a}}e^{-\frac{1}{b-x}}$ is $a<x<b$. Show that $f$ is infinitely differentiable on $\mathbb R$ If $x\neq a,b$ then ...
Ain't No O's user avatar
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0 answers
48 views

Indeterminacy of 1^(infinity)

I am being taught that 1^(infinity) is indeterminate. But I couldn't stop thinking why 1x1x1x1....=1 is incorrect. Please tell me if my reasoning is correct? I thought like this. So, for limit to ...
anuragpsarmah's user avatar
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88 views

How can I rewrite a function so it’s no longer in indeterminate form when $x$ is in the exponent?

I have to rewrite $f(x)$ so that it is no longer is indeterminante as the limit approaches infinity. $$f(x) = \frac{6+2e^{-5x}}{5-16e^{-5x}}$$ I am really not sure how to go about it, though. Can ...
paperJane's user avatar
8 votes
4 answers
425 views

Difficult limit question involving Euler's number and L'Hospital

The limit I'm trying to evaluate is $$ \lim_{x\to+\infty} e^x \left[e - \left(1 + \frac{1}{x}\right)^x\right] $$ After some hours trying, I've made almost no progress. I always end up in some ...
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20 votes
1 answer
337 views

Does $\frac{\sin(tx)}{\sin(x)}$ have a name?

Does the following function have a name? $$\operatorname{boxySine}(t,x) = \begin{cases} \frac{\sin(tx)}{\sin(x)} & x \neq 0 \\ t & \text{otherwise} \end{cases} $$ It appears in ...
Cirdec's user avatar
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1 answer
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Doubt regarding L' Hospital rule

I had calculated $\lim_{z\to 1} (1-z).\tan(πz/2)$ by L'Hospital Rule as Follows I took $\tan(πz/2)$ to the denominator of the denominator i.e $(1-z)/1/\tan(πz/2)$ which becomes $(1-z)/\cot(πz/2)$ and ...
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0 votes
1 answer
224 views

How is Taylor's Theorem Related to Maxima Minima and Indeterminate Forms ( Relationship between them )

I'm reading a real Analysis book in which Taylor's Theorem was introduced in chapter 10 and now the next two chapters are 11. Maxima and Mamina and 12. Indeterminate Forms. In both the chapters the ...
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4 votes
3 answers
341 views

Applying theorems and properties to solve indeterminate limits

I have been trying to solve the following limit for my university Mathematical Analysis class: $$\lim \limits_{x \to +\infty}\sin(x)(\ln{(\sqrt{x}+1})-\ln(\sqrt{x+1}))$$ I know that the answer is 0, ...
Alias's user avatar
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2 votes
2 answers
80 views

Useful theorems/properties for evaluating $\lim \limits_{x \to +\infty}\sqrt{x}(\sqrt{x+1}-\sqrt{x-1})$

I have been trying to solve the following question from my Mathematical Analysis course in university: $$\lim \limits_{x \to +\infty}\sqrt{x}(\sqrt{x+1}-\sqrt{x-1})$$ I am aware that the answer is 1, ...
Alias's user avatar
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0 votes
0 answers
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Limit with an improper integral

Is it true that $$\displaystyle\lim_{x\to 0}x\int_{0}^\infty k^5 dk=0$$ Seems like this has an indeterminate form, but I am being told it equals zero.
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7 votes
1 answer
85 views

Solution verification: Usage of L'Hôpital's rule, derivatives of trigonometric functions

I did the following problem: Find $\displaystyle\lim_{x\to 0} \frac{\cos^2x-1}{x^2}$ The following solution was given: $$\lim_{x\to 0} \frac{\cos^2x-1}{x^2} = \lim_{x\to 0} \frac{2 \sin x \cdot \cos x}...
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