# Questions tagged [limits-without-lhopital]

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

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### Evaluate $\lim_{x \to π/4} \frac{\sqrt{2}- \cos(x)- \sin(x)}{(4x-π)^2}$ [closed]

$$\displaystyle \lim_{x \to π/4} \frac{\sqrt{2}- \cos(x)- \sin(x)}{(4x-π)^2}$$ Can anyone pls help me to figure out the solution without using L'Hopital's rule as I thought if there was an alternative ...
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### What even is the notion of a function growing faster than one another? What defines the fastness (or degree) of a function's growth?

The Context: I was reading the book The Road to Reality by Roger Penrose when in his calculus section he gave a paragraph or two about the notion of $C^{\infty}$-smoothness and there was one exercise ...
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### Why am I getting two different answers using different methods?

I was doing this question: $\lim_{x \to 0} \frac{-\log(1+2h)+2\log(1+h)}{h^2}$ Now using L'Hopital's Rule I was easily able to arrive at the answer which is 1. But when I used the formula for limits ...
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### What's wrong with my solution of James stewart 1.7 Q 36?

Question - Prove that $\lim\limits_{x \to 2} \frac{1}{x} = \frac{1}{2}$ My Solution $|x - 2| < \delta$ $|\frac{1}{x} - \frac{1}{2}| < \epsilon$ $\frac{\delta}{|2(\delta+2)|}<\epsilon\qquad$ ...
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### Conflicting answers when calculating limits [duplicate]

We are asked to evaluate the limit $$\lim_{x \rightarrow \infty}\frac{e^x}{{\left(1+\frac1x\right)}^{x^2}}$$ Applying L'Hospital's rule, we get the correct answer to be $\sqrt e$. However if we apply ...
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### Applying Derivative Formula for $\frac{d}{dx}\frac{1}{\tan x}$

Using the identity, $\cot x = \frac{\cos x}{\sin x}$, I used the derivative formula for limits. This is done as followed: \begin{align} \bigg(\frac{\cos x}{\sin x}\bigg)^{'} &= \lim_{h \to 0} \...
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### Limit of function $\frac{\sin^2(x) - \tan^2(x)}{x^n}$ as $x \to0$

The problem is to find $n$ such that: $$\lim_{x\to 0} \frac{\sin^2(x) - \tan^2(x)}{x^n}$$ is a non-zero real number. My attempt: The limit is of the type $\left[\frac{0}{0}\right]$. By using L'...
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### Noncircular proofs that $\lim_{\theta \to 0} \frac{\sin(\theta)}{\theta}= 1$ [duplicate]

I've been looking over some old calculus stuff, and I came back across the following limit. When deriving $\frac{d}{dx}\sin(x)$, the identity $\lim_{x\to 0}\frac{\sin(x)}{x}=1$ is used as an ...
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### proof of limit without L'Hospital's theorem

I am having trouble to prove these two statements, without L'Hospital's theorem. This came from a series convergence test problem, $\sum_{n=1}^\infty \frac{(\ln n)^3}{n^3}$. If this is proven, then I ...
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### Limit with a geometric interpretation

Let $f:ℝ \to ℝ$ be a $C^∞$ curve. Determine the following limit; $$\lim_{x_1 \to x_2} \dfrac{ \int_{x_1}^{x_2} \sqrt{1+f'(x)^2} dx}{\sqrt{(x_2-x_1)^2+(f(x_2)-f(x_1))^2}}$$ My attempt: I recognized ...
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### Removing Intermediate forms using Integration [duplicate]

As we know that using L'hospital rule rule we can differentiate the numerator and denominator separately to remove the Intermediate form. For example $$\lim_{x\to 0}{\frac{e^x-1}{x^2+x}}$$ So my doubt ...
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### Finding the limit of an equation involving the Gaussian?

Here's a math competition problem that I've been stumped on I tried using L-Hopital's rule, I see how we get the equivalence via the M.V.T., but that approach doesn't seem to be working at all. I'm ...
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### Proving $\lim_{x\to\infty}xa^{x}=0$ in Elementary Ways

I wish to prove the following limit without using L'Hopital rule or other known limits: $$\lim_{x\to\infty}xa^{x}=0$$ where $0<a<1$. I wanted to do so using this sequence limit (which I know how ...
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### Is this a correct approach to calculating $\lim_{n\rightarrow \infty} {\sqrt[n]{\ln(n)}}$?

We have just started covering the limit of sequences and I've stumbled upon this limit in our uni's excercises: $$\lim_{n\rightarrow \infty} {\sqrt[n]{\ln(n)}}$$ I've considered solving it using the ...
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### Can't figure out multivariable limit of $\frac{x^3-x^2y}{x^2+y^6}$ with polar coordinate sub.
I need to find the limit of a function $f(x,y)$ as $(x,y)\rightarrow (0,0)$. The only method I know of is to consider all paths through $(0,0)$ and do polar coordinate substitution to make it into a ...
### What is $\lim_{x\to 0} (1+x)e^{-\left(\frac{1}{|x|} + \frac{1}{x}\right)}$ [closed]
$\lim_{x\to 0} (1+x)e^{-\left(\frac{1}{|x|} + \frac{1}{x}\right)}$ Obviously L'hopital is inapplicable here. I guess it can be done by saying that $e^{-\infty}$ is almost zero so the limit is zero ...