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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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 ...
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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{...
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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:
$$
\lim_{x\to 0}\left(\frac{1}{\sin(x)} - \...
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Electric Field Due to uniformly charged infinitely long wire
Calculate the field at a point P at distance r from infinitely long wire with charge density $\lambda$
Generally the electric field in this case at point P, distance r from the wire is derived using :...
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Is $0/0$ undefined, an indeterminate form, or both? [duplicate]
I have often read that the expression $0/0$ is an indeterminate form. However, it seems to me that it is also undefined. So, what is it really? Is it undefined, an indeterminate form, or, as I suspect,...
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L'Hospital's Rule of infinity over infinity
I am troubled for understanding the L'Hospital's Rule of $\infty/\infty$ :
$$\lim_{x\rightarrow a}\frac{f(x)}{g(x)}= \lim_{x\rightarrow a}\frac{f^\prime(x)}{g^\prime(x)} \tag{1}$$
where $\lim_{x\...
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Difference between 0+0+0+0...infinite times and 0 multiplied by infinity. [closed]
The measure/size of set I came across that the measure of a set that is formed by intersection of countably infinite disjoint sets of measure 0 each, will be the sum of the measures of all the sets = ...
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On the Divergence of an Improper Integral where the Integrand becomes unbounded in the Neighbourhood of a Point inside the Interval of Integration
I have been having some trouble with the way to determine the convergence/divergence of an improper integral such as this: $$\int_0^a \frac{\arctan{(x)}}{1-x^3}dx,$$ where $a>1.$ It is evident that ...
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Fixed point analysis at the origin that appears to be a source, but is undefined.
I have a 2D system of ODEs where $(x,y) \in [0,1]\times(0,1]$ as follows:
$$ \dot{x} = a\frac{x(1-x)}{y} - bxy $$
$$ \dot{y} = cx(1-y) - dy$$
where $a,b,c,d \in \mathbb{R}$. The system is clearly is ...
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Tricky limit on the open first quadrant: $\lim_{(x,y) \rightarrow (0,0)} -x\log{y} $.
Consider $ f:(0,1) \times (0,1) \rightarrow \mathbb{R} $ where$$ f(x,y)=-x\log{y} $$ I am trying to prove whether $$ \lim_{(x,y) \rightarrow (0,0)}{f(x,y)}=0$$ My current idea is as follows:
Let $ p:(...
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Limits of indeterminate forms $(-\infty)(0)$
Can anybody give me a detailed solution on how
$$\lim_{x \to \infty} -x\left(1-e^{-\frac{1}{1+x}}\right) = -1?$$
I understand that separately the limit of each factor is $(-\infty)(0)$ but I could not ...
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Limit of a indeterminate function is always can be calculated?
In class, a teacher made the following statement:
Whenever the limit of a real function of a real variable gives $\frac{0}{0}$ for $x \to a$, the limit can be calculated (via L'Hopital for example).
I ...
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Advanced methods to explain indeterminate forms
I know this post may be covering a subject that is considered 'low quality' but I wanted to try and cover it in a more advanced manner (before writing I also searched if there were duplicate posts).
I ...
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Why $ 0^{\infty} $ isn't indeterminate form?
I was trying to calculate $$
\lim _{x \rightarrow 0} x^{\frac{1}{x}}
$$ I know left hand limit is not equal to right hand limit, hence limit doesn't exist. But I was trying to get their values as well....
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Find the $\lim_{x\to 1} x^{\frac{1}{x^2-1}}$ without applying L'Hôpital's rule
Professor wants me to find the limit
$$
\lim_{x\to 1} x^{\frac{1}{x^2-1}}
$$
without using L'Hôpital and even gave the following advice:
Rewrite the exponent as a product of sum and difference, make ...
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Find Limit : $\lim_{x\to-∞}\frac {1}{xe^{x}}$
If I'm doing everything correctly, I get to 1/(-∞×0) where -∞×0 is undefined, so I don't know what to do. Also, I cannot do a series expansion because x is approaching infinity.
Can someone please ...
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Can indeterminate form be used as a critical point?
If we have a derivative, such as $(e^x-1)/x$, could we test the value of $x$ at $0$ to see it it’s a relative minimum or maximum (using the first derivative test) at that point? In other words, is it ...
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L'Hospital rule for $\frac{0}{0}\cdot\frac{a}{0}$ [closed]
Is there a solution to use the rule of L'Hospital for a boundary value problem like
$$\lim_{x\to1}\left(\dfrac{x^{2}-1}{x-1}\cdot\dfrac{a}{x^{3}-1}\right)$$
where $a> 0$ ?
I know how to solve $\...
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How do you solve $\lim_{n\to\infty}\left(\frac{\log(n+1)}{\log(n)}\right)^{n\log(n)}$ [closed]
$\lim\limits_{n\to\infty}\left(\dfrac{\log(n+1)}{\log(n)}\right)^{n\log(n)}$
As it’s an indetermination, I’ve tried to do
$\exp\left(\lim\limits_{n\to\infty}(n\log(n))\cdot(\log(n+1)/\log(n)-1)\right)$...
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About limits of a particular form
I was talking with some colleagues about why $0^0$ cannot be simply just defined as $1$. Topology came out, set theory came out but most of all a friend of mine said this:
"there is a theorem ...
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Limits using L'Hopital's Rule
I tutor for AP Calc AB, it's been a couple years since I last took calculus though. My student had a question with a limit using L'Hopital's Rule, with $\infty/\infty$.
The question was: $$\lim _{x \...
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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 ...
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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 ...
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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 ...
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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 ...
user1089335
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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 ...
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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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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 ...
user1074209
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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, ...
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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, ...
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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.
user1063886
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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:
$$\lim_{x\to 0} \frac{\cos^2x-1}{x^2} = \lim_{x\to 0} \frac{2 \sin x \cdot \cos x}...
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How do I rigorously compute $\lim_{x\rightarrow0} a^x$ for $a \in \mathbb{R}$? [closed]
How do I rigorously compute $$\lim_{x\rightarrow0} a^x$$ for $a \in \mathbb{R}$?
I can intuitively and graphically get the answer of $\delta_{a\neq0}$ (Kroenecker delta), and I think also by using the ...
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Evaluating $\lim_{x\to0}\,3^{(1-\sec^220x)/(\sec^210x-1)}$
I want to evaluate
$$\lim_{x\to0}\;3^{(1-\sec^220x)/(\sec^210x-1)}$$
So far these have been my ideas, feel free to correct me:
Find that if directly applied to the function, $x_0$ will cause ...
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Evaluating $\lim_{x \rightarrow 0}\frac{3}{x}\bigg(\frac{1}{\tanh(x)}-\frac{1}{x}\bigg)$ using L'Hôpital's rule
I need to evaluate the following limit (using L'Hôpital's rule).
$$\lim_{x \rightarrow 0}\frac{3}{x}\bigg(\frac{1}{\tanh(x)}-\frac{1}{x}\bigg)$$
I expressed $\tanh(x)$ as $\frac{\sinh(x)}{\cosh(x)}$:
$...
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Will the L'Hopital's Rule work here, when it agrees with the derivation of the rule as I know it?
Say I have to find the limit for:
$$\lim_{x\rightarrow a}\frac{f(x)}{g(x)}$$
Such that $$g(a)=0≠f(a)$$
Multiplying the numerator and denominator by $g(x)$, I get$$\lim_{x\rightarrow a}\frac{f(x)g(x)}{...
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Solving the limit $\lim_{n\ \rightarrow\infty}((n+2)!^{\frac{1}{n+2}}-(n)!^{\frac{1}{n}})$ given that it exists finitely
I stumbled across this problem and couldn't solve it.
$$\lim_{n\ \rightarrow\infty}((n+2)!^{\frac{1}{n+2}}-(n)!^{\frac{1}{n}})$$
I tried using different expansions derived by taylor's series and tried ...
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How can I evaluate the limit with $\alpha$ and $\beta$ tend to 1?
I have the equation $F(s,t)=\cfrac{(s\alpha-\beta)e^{-\lambda t}+\beta(1-s)}{(s\alpha-\beta)e^{-\lambda t}+\alpha(1-s)}.$
such that $\lambda = \alpha - \beta$
I need to take the limit as $\alpha$ and $...
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Explaining indeterminate forms
A student of mine asked me why $+\infty -\infty$ and $\frac{\infty}{\infty}$ are considered indeterminate forms.
I think that this is an only apparently stupid question. In other words how can I show, ...
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Limits Question of (0/0) Form
How do I solve this question? Please explain if possible how the method striked you too.
$$\lim _{x\to 0}\left(\frac{\sqrt{p+x}-\sqrt{p-x}}{\sqrt{q+x}-\sqrt{q-x}}\right)$$
I tried double rationalising ...
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Indeterminate and 2 as answer to $\frac {x^2-1}{x-1}$ at x = 1
f(x) = $\frac {x^2-1}{x-1}$
When we directly put x=1 to $\frac {x^2-1}{x-1}$ without simplifying we get $\frac{0}{0}$ indeterminate form but on simplifying further $\frac {x^2-1}{x-1} = \frac{(x-1)(x+...
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Is it valid to apply L'Hospital's rule rule to find $\lim_{x\to0}\frac{(1+x)^{1/x}-e}{x}$?
Consider $\lim_{x\to0}\frac{(1+x)^{1/x}-e}{x}$.
At first glance, it seems like the given limit is not in $\frac{0}{0}$ or $\frac{\infty}{\infty}$ form. (It is $\frac{1^{\infty}-e}{0}$ form.) But I ...
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Indeterminate power direct substitution
Suppose that we have $\lim_{x\to a}f(x)^{g(x)}$ with $f(a)=0$ and $g(a)=0$.
Can we say, that the value of the above limit is $0^0=1$?
Is there a counterexample?
UPDATE: I already saw examples you've ...
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How can I solve limit problems without graphing [closed]
If we have$$\lim_{x\to0^+}\frac{1}{1-e^x} = -\infty$$
What is a good way to solve this without the need to graph the function
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Clarification about $0^0$ [duplicate]
One of the seven indeterminate forms in mathematics is $0^0$ (Refer :) but as a convention its value is taken as $1$ (Refer).
How to resolve this ambiguity ? Any help is much appreciated.
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What exactly does it mean that a limit is indeterminate like in 0/0? [duplicate]
Non-mathematician here (obviously...)
Given this limit question:
$ \lim_{x\to2} \frac{x^2-3x+2}{x^2-4} $
This will have the form $\frac{0}{0}$, which is indeterminate.
This is because both the ...
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An indeterminate expression when calculating derivative.
Good afternoon,
f(x) = x * |cos(pi/x)|
I want to know the value of df/dx (2/3) (right and left hand derivatives)
With a standard algorithm, I calculate it this way:
d/dx x * |cos(pi/x)| = 1 * |cos(pi/...
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An explanation for the result of the following limit
Whilst having troubles in calculating the following limit, which I thought it were indeterminate, I decided to put it into W. Mathematica (the serious software, not W. Alpha online) and it returned ...
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What is a hypergraph consisting of one edge leading from itself to itself?
Is this indeterminate? Undefined? Meaningless?
I became confused when I looked at it by starting with
$$(A \to B) \to (A \to B). \qquad\label{1}(1)$$
Since both ends of this edge are this edge, it's ...
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factor by the leading term to compute a limit depending on $\pm \infty$
I have to compute the limit of $ f(x)=\frac{2+x+e^x}{2+x-e^x}$ when $x$ approaches $\pm \infty$.
My question is about the leading term. Am I right if I consider that $e^x$ is my leading term when $x \...
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