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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Solution verification: Usage of L'Hopitâl'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 $a \in \mathbb{R}$?

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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Proof of power rule for negative integer exponents involving indeterminate forms

Proof of the power rule for negative integer exponents Theorem statement: $$z \in \Bbb Z^-, \ \ f(x) = cx^z \implies f'(x) = czx^{z-1}$$ Proof: $$f'(x) = \lim_{d \rightarrow 0} \frac{c/(x+d)^{|z|} - ...
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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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2 votes
1 answer
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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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2 answers
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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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3 answers
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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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1 answer
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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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Why is the limit of $\frac{1}{1-e^x}$ when $x$ approaches $0^+$ equal to $-\infty$? [closed]

$$\lim_{x\to0^+}\frac{1}{1-e^x} = -\infty$$ Why is the limit equal to $-\infty$ and not $\infty$? Isn't $e^x$ at $x=0$ equal to $1$? Thus, $\frac{1}{1-1}=\infty$.
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1 answer
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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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1 answer
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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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1 answer
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Why ${(0^+)}^{+ \infty}$ is well defined and not ${(0^+)}^{- \infty}$?

I want to compute the limit of $|x|^{\frac{1}{x}}$ when $x$ tends to $0$. When $x$ tends to $0$ from above, I get ${(0^+)}^{\frac{1}{0^+}}= {(0^+)}^{+\infty}=0$. However when, I approach $0$ from ...
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2 answers
110 views

Whenever we get undefined or indeterminate expressions in the real world, do we just take the limit, if it exists?

In the real world, that is, in engineering, computer science, or whatever, whenever we get undefined or indeterminate expressions, do we just take the limit, if it exists? Does this work in the real ...
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1 vote
1 answer
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Limit of a function with 2 variables of the indeterminate form 0/0

For my vector calculus course, I need to solve this limit: $$\lim_{(x,y)\rightarrow (0,0)} \frac{1-\cos(x^2y)}{x^6+y^4}$$ By replacing (x,y) by (0,0), I noticed that the limit was of the form $\frac{0}...
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$\lim_{s\to1}\frac{t-s}{1-s}$ when $0\leq s\leq t\leq 1$

I was working on a topology exercise and got to a point (no pun intended) where I had to analyze the behaviour of $f(t,s)=\frac{t-s}{1-s}$ as s goes to $1$, under the assumption that $t\geq s$. It ...
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2 answers
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Exercise on limit of indeterminate form: $\lim_{x \rightarrow 1} \frac{\sqrt{x+3} - \sqrt{5-x}}{\sqrt{1+x} - \sqrt{2}} = \sqrt2$

I'm not able to show that $\lim_{x \rightarrow 1} \frac{\sqrt{x+3} - \sqrt{5-x}}{\sqrt{1+x} - \sqrt{2}} = \sqrt{2}$ I proceeded as follows: Substituting 1 to $x$ gives the indeterminate form $\frac{0}{...
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1 answer
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Evaluate the limit $\lim_{x \to 0}\frac{2+\tan \left(e^{x}-\cos x\right)-e^{x}-\cosh x}{x(\sqrt{1+2 x}-\sqrt[3]{1+3 x})}$

Evaluate the limit $$L=\lim _{x \to 0}\frac{2+\tan \left(e^{x}-\cos x\right)-e^{x}-\cos h x}{x(\sqrt{1+2 x}-\sqrt[3]{1+3 x})}$$ By generalized binomial expansion we have $$\sqrt{1+2 x}-\sqrt[3]{1+3 x}=...
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1 answer
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symbol isn't changing for product rule

I was doing Indeterminate Form. I saw something weird while differentiating using product rule. $$f(x)g(x)=f'(x)g(x)+f(x)g'(x)$$ I got a value which was looking like. $$\lim_{x->0} \frac{e^x-e^{\...
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0 answers
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How to solve an indecision form of the type 0/0 of a ratio of integrals?

It's a couple of days that i am stuck with the following problem. Consider the following limit of ratio of integrals: $$ \lim_{p\to\infty}\;\frac{\int p(y_i|\mu)^2\cdot p(\mu)d\mu} {\left(\int p(y_i|\...
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0 answers
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Why is $1^{\infty}$ indeterminate? [duplicate]

I've learned indeterminate forms and l'Hospital's Rule recently. The text book says that $1^{\infty}$ is an indeterminate form. Why?? In my understanding, $1^{\infty}$ is the product of infinite ...
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Integrating an inverse natural log function

I am currently trying to find the primitive of the following function: $$f(t)= \int \frac{a}{\ln(at+b)}dt$$ Where $a$ and $b$ are constants. Furthermore $a>0$ and $0<b<1.$ Additionally, $a$ ...
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4 answers
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Evaluating $\lim_{n\to\infty}\left(1 + \frac{\sin n}{5n + 1}\right)^{2n + 3}$ (a $1^\infty$ indeterminate form)

My professor gave us this limit as part of homework: $$ \lim_{n\to\infty}\left(1 + \frac{\sin n}{5n + 1}\right)^{2n + 3} $$ I can see it's in the indeterminate form $1^{\infty}$, so my first thought ...
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1 vote
1 answer
88 views

Indeterminate limits involving square roots

So, I've been for the past $2$ days trying to solve these two limits: $$\lim_{x \to 4} \frac{3 - \sqrt{5 + x}}{1 - \sqrt{5 - x}}$$ $$\lim_{x \to 2} \frac{x^2 + 3x - 10}{3x^2 - 5x - 2}.$$ My problem ...
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2 answers
33 views

How is the limit of this function calculated

The function $f$ is defined as: $$f(a) = \frac{\sigma}{\sqrt{2a}} \sqrt{1-e^{-2as}} Q_p$$ I want to calculate the limit $ lim_{a\to0} f(a)$. The function is in indeterminate $\frac00$ form. The answer ...
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0 votes
3 answers
180 views

L'Hospital's Rule for the indeterminate form $0^0$

I'm teaching an introductory real analysis course for the first time this year and one of the textbook problems asks to prove the following: Let $f$ be differentiable on $(a,b)$ and let $c \in (a,b)$....
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2 answers
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Evaluating limits approaching infinity

How to calculate the limit of this function? $$f(x) = \frac{(-1)^x \sqrt{x-1}}{x},\ x = \{1, 2, 3, 4...\}$$ So, I have tried calculating the limits way: First, by using the multiplication rule; $$\...
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5 votes
1 answer
81 views

Does the limit $\lim _{x \rightarrow 0} \frac{\int_{0}^{x^{2}} \sin \sqrt{t} d t}{x^{3}}$ exists?

Evaluate $$L=\lim _{x \rightarrow 0} \frac{\int_{0}^{x^{2}} \sin \sqrt{t} d t}{x^{3}}$$ Since the limit is $\frac{0}{0}$ form, Using L'Hopital's rule and Leibniz rule we get $$L=\lim_{x \to 0}\frac{2x\...
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1 vote
0 answers
31 views

Tricky Indeterminate Power With Indeterminate Solution

I have learn in L'hospital rule about indeterminate power, we will take $\ln$ to find the limit. But in this question, I suspect there is no solution to the limit. $\lim_{x\to \infty}({\frac {2-3x}{1+...
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1 vote
4 answers
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$\lim_{x\to \infty} (x^4 +5x^3+3)^c-x=k$. Find $k$ and $c$.

$\lim_{x\to \infty} (x^4 +5x^3+3)^c-x=k$. Find $k$ and $c$. key points: This is supposed to be solved with knowledge of just L'hospital's rule. *The value of limit i.e, k is finite and non zero I ...
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1 answer
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calculating $\lim\limits _{x\to0}\left(\frac{1}{x^{2}}-\frac{e^{x}}{\left(e^{x}-1\right)^{2}}\right)$ using taylor polynomials

I am trying to calculate the limit $$\lim\limits _{x\to0}\left(\frac{1}{x^{2}}-\frac{e^{x}}{\left(e^{x}-1\right)^{2}}\right)$$ using taylor polynomials. I tried using L'Hôpital's rule, but it was ...
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0 votes
2 answers
94 views

Why is $0/\lim_{x\to0} x$ undefined, when $\lim_{x\to2} (x^2-4)/(x-2)$ is defined?

For $\lim\limits_{x\to2}\ (x^2-4)/(x-2)$, we are able to cancel out $x-2$ and rewrite it as $\lim\limits_{x\to2} x+2 $ But in maths, we are not able to cancel out $0$ values so $x-2$ is not zero, and ...
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1 vote
2 answers
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$0^\infty$ and indeterminate forms

I have actually two related questions: 1. Should indeterminate forms be able to attain an infinite number of values to be considered indeterminate? I’m asking because Wikipedia says: The expression 1/...
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0 votes
1 answer
60 views

How to prove that a form like $0^\infty$ is not an indeterminate form? What makes a form indeterminate?

I just wanted to know why some particular forms (seven of them) are called indeterminate forms. Why are there only seven indeterminate forms? Can anyone please prove why $0^\infty$ is not an ...
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2 votes
2 answers
41 views

Can we apply L'hopital Rule when we have $\frac{-\infty}{+\infty}$?

I saw in a book, it evaluated $\lim_{x\to0^+}\cfrac{(\ln x)^3}{\frac1x}$ with L'hopital Rule. the fraction is $\frac{-\infty}{+\infty}$ actually. so can I use L'hopital rule safely every time I see ...
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  • 5,575
2 votes
2 answers
185 views

Infinity as a limit in an indeterminate form

There is this limit $$\lim_{x\to1}\frac{\ln x}{\left(x-1\right)^{2}}$$ for which I built a graph and I know the answer, so the question is not about how to compute it, but about my observation that ...
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2 votes
1 answer
99 views

Is it logical to say that $ 2\over 0$ $\ne$ $ 2\over 0$?

As I was doing a math exercice, I came across a question which I decided to prove by contrapositive. That required me to show that $ f(4-x)$ $\ne$ $f(x)$ - but in both cases the result was $ 2\over 0$....
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4 votes
3 answers
111 views

Restrictions on laws

I'm wondering about the restrictions, my doubt is for example at $\log_a(b)=c\implies a^c=b$, how would anyone add the restrictions for this? I know the argument and the base of a log have to be >0 ...
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3 votes
3 answers
417 views

Why is $\lim_{x \to \infty} x \log(1+\frac{1}{x}) = \lim_{y \to 0} \frac{\log(1+y)}{y}$?

Why is $$\lim_{x \to \infty} x \log(1+\frac{1}{x}) = \lim_{y \to 0} \frac{\log(1+y)}{y} ?$$ I understand that they are both indeterminate forms. Specifically we are initially given $$\lim_{x \to \...
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  • 5,345
1 vote
1 answer
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Prove identity of indiscernibles

Define the earth mover's distance as \begin{equation} EMD(\mathbf{x},\mathbf{y}) = \frac{\sum_{i=1}^m\sum_{j=1}^n f_{ij}d(x_i,y_j)}{\sum_{i=1}^m\sum_{j=1}^n f_{ij}}. \end{equation} where we define ...
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1 vote
6 answers
317 views

Undetermined or indeterminate forms: $\frac{0}{0}, \frac{\infty}{\infty}, 0\cdot\infty, 1^\infty, 0^0, +\infty-\infty$

I wanted to know who has decided that for the calculation of the limits of the following forms, $$\color{orange}{\frac{0}{0},\quad \frac{\infty}{\infty},\quad 0\cdot\infty,\quad 1^\infty,\quad 0^0,\...
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  • 4,858
5 votes
3 answers
185 views

verifying $\lim\limits_{x \to \infty} (1+\frac{1}{\sqrt{x}})^x$

What is the limit of $\lim\limits_{x \to \infty} (1+\frac{1}{\sqrt{x}})^x$ ? I tried to solve it but I am not sure if it is appropriate to solve it this way. $(1+\frac{1}{\sqrt{x}})^x =\exp\Bigl({x\...
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