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.

learn more… | top users | synonyms

4
votes
3answers
255 views

The Google calculator says that $\left(\frac00\right)^0=1$. Is this true?

According to Google, $\left(\frac00\right)^0=1$. Is this true? Why or why not?
13
votes
11answers
1k views

Why does simplifying a function give it another limit [duplicate]

I'm asked: $$\lim_{x\to 1} \frac{x^3 - 1}{x^2 + 2x -3}$$ This does obviously not evaluate since the denominator equals $0$. The solution is to: $$\lim_{x\to 1} \frac{(x-1)(x^2+x+1)}{(x-1)(x+3)}$$ ...
0
votes
2answers
35 views

How to find slope at a point where the derivative is indeterminate

How should I find the slope of a curve at origin whose derivative at the origin is indeterminate. My original problem is to calculate the equation of tangent to a curve at origin. But for the equation ...
0
votes
0answers
8 views

How do I integrate a function with a continuous domain where the integrand is indeterminate ($0.\infty$)?

I have an integral to solve: $$I=\int f\left(g,h\right)~dg = \int_0^1 \theta\left(g-h\right)\log_e\frac{1-h}{\theta\left(g-h\right)}~dg$$ where $\theta()$ is the Heaviside step function, and $h$ is ...
2
votes
1answer
64 views

Exponential limit of the form $0^\infty$

I was trying to derive a general expression for the limit $$\large{y=\lim_{x\to a} f(x)^{g(x)}}$$ where $\lim_{ x \to a} f(x)=0$ and $\lim_{ x \to a} g(x)=\infty$ $$$$ I managed to reach till here: $$\...
0
votes
2answers
62 views

Calculate $\lim_{n \to \infty } \frac{1}{n^2} \int_0^n \frac{ \sqrt{n^2-x^2} }{2+x^{-x}} dx$

I need to calculate the limit$$\lim_{n \to \infty } \frac{1}{n^2} \int_0^n \frac{ \sqrt{n^2-x^2} }{2+x^{-x}} dx$$ How could I calculate this? Any hlep would be appreciated.
1
vote
1answer
20 views

$\lim_{x\to 0}x^a\log^k(x)$ where $a>0,k\in\mathbb N_0$

I'd know how to solve this for $k=0$ or $k=1$ for example, but I'm currently lost trying to prove the limit is zero for any non-negative integer $k$. I'd appreciate any hints!
5
votes
2answers
188 views

Limit with x approaching infinity [closed]

The problem says: If $$\lim_{x\to +\infty} \left\lbrack\frac{ax+1}{ax-1}\right\rbrack^x=9$$, determine $a$. It appears to be a case of $\left\lbrack\frac{\infty}{\infty}\right\rbrack^\infty$. ...
0
votes
2answers
34 views

the limit of a sequence to another sequence indeterminate form

I am trying to solve this limit question...I have tried to take the natural log of an^(bn), but somehow ended up with infinity times infinity. Is there any way to this? Thanks a lot.
0
votes
3answers
53 views

Limit Calculation $\lim_{x\to \infty} x\log(x^2+x)- x^2\log(x +1)$

I am having a problem with the calculation of the following limit. I need to find $$\lim_{x\to \infty} x\log(x^2+x)- x^2\log(x +1).$$ I've been trying in this way but I'm not sure if it is correct: ...
0
votes
2answers
42 views

Kind of limit I don't know how to solve

Suppose that we have three functions with the following behavior: $$\lim_{x\to x_0} f(x)=0$$ $$\lim_{x\to x_0} g(x)=0$$ $$\lim_{x\to x_0} h(x)=\infty$$ I don't know how to approach a limit of the ...
2
votes
3answers
55 views

Query regarding other seemingly indeterminate forms

I know there are 7 indeterminate forms as follows- $$0^0$$ $$1^{\infty}$$ $${\infty}^0$$ $$\frac{0}{0}$$ $$\frac{\infty}{\infty}$$ $$0\cdot\infty$$ $${\infty}-{\infty}$$ I cant help but wonder if ...
3
votes
2answers
41 views

Finding the limit as $n \to \infty $ of $n\ln\left(1+\frac{\ x}{n^2}\right)$

Find $$\lim_{n\to \infty} n\ln\left(1+\frac{\ x}{n^2}\right)$$ My attempt: $\lim_{n\to \infty} n \left[\ln\left(\frac{\ n^2 +x}{n^2}\right)\right]$ = $\lim_{n\to \infty} n [\ln (n^2 +x) - \...
1
vote
5answers
90 views

Limit of $x - \ln(x)$ as $x$ approaches $+\infty$

To evaluate the limit of an even larger expression $$ \lim_{x \to +\infty} \frac{\ln(\ln x)}{\ln(x - \ln x)} $$ I need to evaluate part of the denominator to determine whether I could apply L'...
0
votes
2answers
33 views

Limits which involve an explicit $0$

I am trying to solve the following limit (or prove it doesn't exist) $$ \lim_{(x,y) \rightarrow (0,0)} \frac{x^2}{||x,y||} $$ where $(x, y) \in R^2$. I decided to analyze the limit over the y-axis, ...
2
votes
2answers
63 views

What's the limit of $\lim_{x\to -1} \frac{x^{101}+1}{x+1} $

I've tried a lot in how to solve this limit, What's is the limit of $$\lim_{x\to -1} \frac{x^{101}+1}{x+1} $$
3
votes
4answers
99 views

Give examples showing why $0\cdot \infty$, $\infty/\infty$, and $0/0$ are meaningless

Assuming arithmetic operations on $\overline{\mathbb{C}}$ (that's the extended complex plane) are defined via arithmetic operations on the corresponding sequences, I need to give examples showing why $...
0
votes
1answer
20 views

Indeterminate form and equality of expressions.

Given this chain of equalities: $$\frac{\sqrt{1-x} - \sqrt{1+x}}{x}=\frac{(\sqrt{1-x} - \sqrt{1+x})(\sqrt{1-x} + \sqrt{1+x})}{x(\sqrt{1-x} + \sqrt{1+x})} = \frac{(\sqrt{1-x})^2 - (\sqrt{1+x})^2}{x(\...
1
vote
0answers
47 views

Existence and non-singularity of the Fisher information matrix

Consider a random vector $X$ defined on the probability space $(\Omega, \mathcal{F}, \mathbb{P})$, $X: \Omega \rightarrow \mathbb{R}^k$. Suppose $X$ has probability density $p_{\theta_0}$ with respect ...
3
votes
7answers
261 views

Limit $\lim_{x \to 0} \dfrac{\sin(1-\cos x)}{ x} $

$$\lim_{x \to 0} \dfrac{\sin(1-\cos x)}{ x} $$ What is wrong with this argument: as $x$ approaches zero, both $x$ and $(1-\cos x)$ approaches $0$. So the limit is $1$ . How can we prove that they ...
1
vote
6answers
153 views

“Proving” that $0^0 = 1$ [duplicate]

I know that $0^0$ is one of the seven common indeterminate forms of limits, and I found on wikipedia two very simple examples in which one limit equates to 1, and the other to 0. I also saw here: ...
0
votes
1answer
24 views

Does symmetric property of Equivalence hold when the left hand side of the equation is in indeterminate form?

The Symmetric Property of Equivalence is a=b implies b=a. This property does not have any conditions on a or b. But what if through manipulation I get $a=\frac00$ where $a$ is a real number does this ...
-1
votes
2answers
75 views

Need help solving these limits [closed]

$$1. \lim_{n \to \infty} \left(\frac{2n^2 + 5}{1 + 5 + ... + (4n - 3)} + \frac{5}{n}\right)$$ $$2. \lim_{x \to -1} \left(\frac{x^4 + 2x^3 + 4x^2 + 6x + 3}{x^3 - 3x^2 - 9x - 5}\right)$$ $$3. \lim_{x \...
0
votes
1answer
63 views

List of Indeterminate forms in Mathematics

I know that , $1) \frac{0}{0}$ $2) \frac{\pm\infty}{\pm\infty}$ $4) 0 \times(\pm\infty) $ are Indeterminate forms. But in measure theory $ 0 \times(\pm\infty) =0 $ Are there any other ...
3
votes
7answers
315 views

Does $1^{\infty}=e$ or $1^{\infty}=1$?

In fact the real question is: Does $\lim\limits_{n\to\infty}1^{n}=e$?. I know that $$ \lim\limits_{n\to\infty}\left(1+\dfrac{1}{n}\right)^n=e, $$ So, can we say that $1^\infty=e$? And, by logic, ...
0
votes
1answer
31 views

Plotting a function around $0$ shows it is jumping around, although the limit as $x\to 0$ exists

My problem revolves around the function: $$ f(x) = \frac{\sin(\tan x) - \tan(\sin x)} {\arcsin(\arctan x) - \arctan(\arcsin x)} $$ The $\lim_{x\to 0}f(x)= 1$. However, whilst approaching 0, there is ...
3
votes
1answer
58 views

How do you prove indeterminate form using epsilon and delta?

The question, for instance, is proving $$\lim_{x\to\infty}\frac{x}{x+1}=1$$ This is my answer, which is likely incorrect. $$\forall\epsilon>0, \exists M \in \mathbb{R}$$ such that $$ x>M \...
1
vote
3answers
43 views

Evaluating the limit $\mathop {\lim }\limits_{x \to 0} \frac{{x(1 - 0.5\cos x) - 0.5\sin x}}{{{x^3}}}$

For evaluating the limit $\lim\limits_{x \to 0} \frac{x(1 - 0.5\cos x) - 0.5\sin x}{x^3}$, I proceeded as follows: $$\lim_{x \to 0} \left(\frac{x(1 - 0.5\cos x)}{x^3} - \frac{0.5}{x^2}\left(\frac{\...
0
votes
0answers
32 views

How to solve a particular indeterminate form

So the answer says $$\lim_{x\to \infty}x^2\sin\left(\frac1x\right)=\lim_{h\to 0^+}\frac1h\frac{\sin h}h$$ How does the transformation work?
12
votes
7answers
908 views

Question about the derivative definition

The derivative at a point $x$ is defined as: $\lim\limits_{h\to0} \frac{f(x+h) - f(x)}h$ But if $h\to0$, wouldn't that mean: $\frac{f(x+0) - f(x)}0 = \frac0{0}$ which is undefined?
4
votes
2answers
262 views

What's the value of $\frac{(x-1)x^2}{x-1}$ when $x=1$?

Consider the function $$f(x)= \frac{(x-1)x^2}{x-1},\quad x \in \Bbb R\ .$$ This function gives simply gives $f(x)=x^2$ by cancelling the term $x-1$, if I am not wrong. The variable here is $x$ which ...
3
votes
2answers
61 views

Prove that $\lim_{x\to 0}\frac{1}{\ln(1+x)}-\frac{1}{\ln\left(x+\sqrt{1+x^2}\right)}=\frac{1}{2}$ [duplicate]

Prove that $$\lim_{x\to 0}\frac{1}{\ln(1+x)}-\frac{1}{\ln\left(x+\sqrt{1+x^2}\right)}=\frac{1}{2}$$ I can solve this problem using L Hospital Rule by converting it into $\frac{0}{0}$ form. But the ...
1
vote
5answers
84 views

How do I evaluate this limit :$\displaystyle \lim_{x \to \infty} ({x\sin \frac{1}{x} })^{1-x}$.?

I would like to evaluate this limit :$$\displaystyle \lim_{x \to \infty} ({x\sin \frac{1}{x} })^{1-x}$$. I used taylor expansion at $y=0$ , where $x$ go to $\infty$ i accrossed this problem : ${1}^{...
0
votes
0answers
61 views

Indeterminate forms of Limits in Multivariable Calculus

In multivariable calculus, I have come across different approaches to operate on indeterminate forms. I have come across cases where you choose different paths to test whether the limit takes the same ...
3
votes
5answers
96 views

How do I solve $\lim_{x \to 0} \frac{\sqrt{1+x}-\sqrt{1-x^2}}{\sqrt{1+x}-1}$ indeterminate limit without the L'hospital rule?

I've been trying to solve this limit without L'Hospital's rule because I don't know how to use derivates yet. So I tried rationalizing the denominator and numerator but it didn't work. $\lim_{x \to 0}...
2
votes
1answer
90 views

If 0 / 0 is indeterminate, are all clauses “0 / 0 != x” true

Elsewhere arose a discussion about logical clauses that can be made from indeterminate forms, in this case, namely $0 / 0$. Since $0 / 0$ is indeterminate form, can we make these logical clauses: $0 ...
1
vote
2answers
41 views

Having trouble with certain indefinite forms of a limit

I have been having a rather difficult time trying to solve this limit $$ \lim_{x\to0}\bigg(\frac{5}{x^4}-\frac{5}{x^2}\bigg) $$ So far, I have rewritten it to this point $$ \lim_{x\to0}\bigg(\frac{-...
0
votes
3answers
118 views

Is $\lim\limits_{x,y\to-\infty}\frac{\sqrt x\sqrt y}{\sqrt{xy}}=1$?

WolframAlpha is suggesting (judging by the plot given) that the limit is actually $-1$. I would think the following manipulations would be okay to conclude that is the opposite. $$\lim_{x,y\to-\infty}...
0
votes
1answer
32 views

Test for convergence - Log operation

∑ Log (n / n+ 1) I solved the above problem in two different ways. 1st Method = Log n - Log n+1 = Log n - Log n . Log 1 = Log n - Log n . 0 = Log n = Log ∞ = ∞ (Diverges) Method 2 = Log ( n / n + ...
1
vote
2answers
43 views

Prove that $\lim_\limits{x\to 1}{\frac{f^2(x)+g^2(x)}{f(x)+g(x)}}=+\infty$

Let $f,g:\mathbb{R}\rightarrow\mathbb{R^*_+}$ be functions such that: $\lim_\limits{x\to 1}{f(x)}=+\infty$ and $\lim_\limits{x\to 1}{g(x)}=+\infty$ Prove that: $$\lim_\limits{x\to 1}{\frac{...
4
votes
2answers
73 views

The indeterminacy of 0/0 and vacuous truth?

Today, my roommate and I picked up our friend from the airport. We were supposed to pick him up yesterday, but he missed his flight. We joked that he misses flights a lot, and that only catches 70% of ...
0
votes
1answer
41 views

Conventions adopted for extended reals

It is known that $0^0$ despite being an indeterminate limit form, is usually defined to be equal to $1$. I wonder whether similar conventions exist for some other "indeterminate forms" in the context ...
11
votes
5answers
1k views

Why does L'Hôpital's rule work for sequences?

Say, for the classic example, $\frac{\log(n)}{n}$, this sequence converges to zero, from applying L'Hôpital's rule. Why does it work in the discrete setting, when the rule is about differentiable ...
2
votes
2answers
195 views

Is $\frac00=\infty$? And what is $\frac10$? Are they same? Does it hold true for any constant $a$ in $\frac{a}0$ [duplicate]

I know that $\lim_{x\to0}\frac{x}{x}=$ 1. But in my text book, it is written that it is $\infty$ and even $\frac10=\infty$. But how is it possible? And are they both same? What is the difference ...
1
vote
3answers
61 views

Why don't we use undefined forms to find limits of infinity? [closed]

I came across a couple of simple limits of infinity and solved them using the normal subsequence method. Yet, I also tried solving them by substituting the undefined form $\frac{1}{0}$, and although ...
0
votes
1answer
54 views

What does it mean for a “formula to be undefined”?

I was covering the techniques used sketch rational functions of five different types as follows: However, then I encountered this: And, Ij just can't find out what it means for the formula to ...
4
votes
3answers
310 views

Infinity indeterminate form that L'Hopital's Rule: $\lim_{x\to0^+}\frac{e^{-\frac{1}{x}}}{x^{2}}$

When I tried to find the limit of $$ \lim_{x\to0^+}\frac{e^{-\frac{1}{x}}}{x^{2}} $$ by applying L'Hopital's Rule the order of denominator would increase. What else can I do for it?
0
votes
4answers
80 views

Undefined reference $1^{\infty}$ in a limit such as $\lim_{n\to\infty}(1+1^n)$ [closed]

If $1^{\infty}$ is undefined reference in a limit, how is $$\lim\limits_{n\to\infty}(1+1^n)=2$$
1
vote
1answer
47 views

Indeterminant in Summation

I have the following summation: $$\sum_{k=0}^\infty(1-e^x)^k=\sum_{k=0}^\infty\sum_{j=0}^k\binom{k}{j}(-1)^je^{jx}$$ Then $$e^{jx}=\sum_{i=0}^\infty j^i\frac{x^i}{i!}$$ So, $$\sum_{k=0}^\infty\sum_{...
0
votes
1answer
27 views

If period of $\sin^{2m} \sqrt{k}x$, $m\in \mathbb N$ is $\pi$ then find $\lim \limits_{n\to \infty }k^{n}$.

If period of $\sin^{2m} \sqrt{k}x$, $m\in \mathbb{N}$ is $\pi$ then find $\lim \limits_{n\to \infty }k^{n}$. My attempt: Period of $$\sin^{2m} \sqrt{k}x=\frac{\pi}{\sqrt{k}}$$ According to ...