Questions tagged [limits-without-lhopital]

The evaluation of limits without the usage of L'Hôpital's rule.

The idea here is to evaluate the limit using standard limit theorems (algebra of limits, Sandwich/Squeeze Theorem, essentially without using any differentiation) and some standard limit formulas related to algebraic, trigonometric, exponential and logarithmic functions. Very often, Taylor series techniques prove fruitful in such problems as they allow for easy cancellation of powers and most terms evaluate to zero, leaving a simple expression for the limit.

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Limits of functions that can't be attacked by Taylor series or L'hopital's rule

Often on this site there is posed a question about a limit which is hard to resolve using l'Hopital. (I don't mean probelms asking to find a limit without using 'Lhopital, I mean problems where using l'Hopital leads to roadblocks or subtleties.) I…
Mark Fischler
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Prove this limit without using these techniques, and for beginner students: $\lim_{x\to0} \frac{e^x-1-x}{x^2}=\frac12$

How can we prove that $$\lim_{x\rightarrow 0}\cfrac{e^x-1-x}{x^2}=\cfrac{1}{2}$$ Without using L'hopital rule, and Taylor expansions? Thanks
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Evaluate $\lim_{x\to 1} \frac{p}{1-x^p}-\frac{q}{1-x^q}$

Evaluate $$\lim_{x\to 1} \frac{p}{1-x^p}-\frac{q}{1-x^q}$$ where $p,q$ are Natural Numbers. I tried rationalization, but I wasn't able to get anywhere. I'm not being able to remove the $\infty-\infty$ indeterminate form. Any help would be…
user342209
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Find $\lim_{x\to \frac\pi2}\frac{\tan2x}{x-\frac\pi2}$ without l'hopital's rule.

I'm required to find $$\lim_{x\to\frac\pi2}\frac{\tan2x}{x-\frac\pi2}$$ without l'hopital's rule. Identity of $\tan2x$ has not worked. Kindly help.
golu
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Finding $\lim_{x\to 0} \frac {2\sin x-\sin 2x}{x-\sin x}$ geometrically

While looking at this question, I noticed an interesting geometric interpretation of the limit the OP was trying to evaluate. His limit came to twice the value of the limit $$\lim_{x\to 0}\frac{\sin x (1-\cos(x))}{x-\sin x}$$ which can be…
user88319
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Solving $\lim\limits_{x\to0} \frac{x - \sin(x)}{x^2}$ without L'Hospital's Rule.

How to solve $\lim\limits_{x\to 0} \frac{x - \sin(x)}{x^2}$ Without L'Hospital's Rule? you can use trigonometric identities and inequalities, but you can't use series or more advanced stuff.
ChessMaster
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What is the limit $\lim_{n\to\infty}\frac{1^n+2^n+3^n+\dots+n^n}{n^{n+1}}$?

I want to determine the limit given below: $$\lim_{n\to\infty}\frac{1^n+2^n+3^n+\ldots+n^n}{n^{n+1}}$$ I have tried to solve thise several times ,but with no results.I have tried using lema stolz cezaro and managed to find a general formula for the…
Razvan Paraschiv
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Limit of $x^2\sin\left(\ln\sqrt{\cos\frac{\pi}{x}}\right)$

Find $$\lim_{x\to\infty}x^2\sin\left(\ln\sqrt{\cos\frac{\pi}{x}}\right).$$ I tried substituting $x=1/t$ with $t$ approaching $0$ but the term inside the bracket is not giving me ideas on how to compute the limit.
user167045
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$\lim_{x\to0} \frac{x-\sin x}{x-\tan x}$ without using L'Hopital

$$\lim_{x\to0} \frac{x-\sin x}{x-\tan x}=?$$ I tried using $\lim\limits_{x\to0} \frac{\sin x}x=1$. But it doesn't work :/
IndyZa
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Solution verification:$\lim_{x\to 2}\frac{\ln(x-1)}{3^{x-2}-5^{-x+2}}$

Evaluate without L'Hospital:$$\lim_{x\to 2}\frac{\ln(x-1)}{3^{x-2}-5^{-x+2}}$$ My attempt: I used: $$\lim_{f(x)\to 0}\frac{\ln(1+f(x))}{f(x)}=1\;\&\;\lim_{f(x)\to 0}\frac{a^{f(x)}-1}{f(x)}=\ln a$$ $$ \begin{split} L &= \lim_{x\to 2}…
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Finding limit for infinite quantities.

Find $$\lim_{x\rightarrow 0}{\color{red}{x}} \cdot\bigg(\dfrac{1}{1+x^4}+\dfrac{1}{1+(2x)^4}+\dfrac{1}{1+(3x)^4}\cdots\bigg)$$ As terms are written infinitely my intuitions doesn't let to give answer as $0$. So tried calculated that weird sum…
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Evaluate $\lim_{n\rightarrow\infty}\sum_{k=1}^n\arcsin(\frac k{n^2})$

Compute $$\lim_{n\to\infty}\sum_{k=1}^n\arcsin\left(\frac k{n^2}\right)$$ Hello, I'm deeply sorry but I don't know how to approach any infinite sum that involves $\arcsin$, so I couldn't do anything to this question. Any hints/tips would be…
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Compute $\lim_{n\rightarrow\infty}\frac{n^n}{(n!)^2}$

I have to compute $\lim_{n\rightarrow\infty}\frac{n^n}{(n!)^2}$. I tried say that this limit exists and it's l, so we have $\lim_{n\rightarrow\infty}\frac{n^n}{(n!)^2} = L$ then I rewrited it as: $\lim_{n\rightarrow\infty}(\frac{\sqrt…
C. Cristi
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Calculating a limit with trigonometric and quadratic function

$$\lim_{n->\infty}\left(1+{1\over{n^2+\cos n}}\right)^{n^2+n}$$ I vaguely get the idea that since $\cos n$ and $n$ dont really matter compared to $n^2$, this must evaluate to $e$. But not sure how to prove this. Further do all the limits of form…
Anvit
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Limit problem: $\lim_{t\to1} \frac {\sqrt {2t^2-1}\sqrt[3]{4t^3-3t}-1}{t^2-1}$

Update: $$\lim_{t\to1} \frac {\sqrt {2t^2-1}\sqrt[3]{4t^3-3t}-1}{t^2-1}$$ First of all, I am grateful to you for all the answers you have given me. I want to ask MSE to confirm the correctness of the alternate solution and its mistake. I worked so…