Mathematics Stack Exchange
Mathematics Stack Exchange

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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Appropriate change of variable for limit (sine function)

I couldn't find this particular solution on here; I apologise in advance if it has been posted before. I know that $\lim_{x \rightarrow 0} \frac{\sin(x)}{x} = 1$ I am asked to find this limit, using the limit above and using a change of variable, so…
user9750060
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I need the steps to solving this limit without using l´Hopital rule

I've tried many ways of solving this limit without using l'Hopital and I just can't figure it out. I know the answer is $3/2 \sin (2a).$ $$\lim_{x\,\to\,0} \frac{\sin(a+x)\sin(a+2x) - \sin^2(a)} x$$ Thank you!
Andreina Ayala
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How to find : $\lim_{x \to \pi/2}\frac{1-\sin x}{\sin x+\sin 3x}$

How to find : $$\lim_{x \to \pi/2}\frac{1-\sin x}{\sin x+\sin 3x}$$ My Try : $$x-\pi/2=t \to x=t+\frac{\pi}{2}$$ And: $$ \sin (t+\frac{\pi}{2})=\cos t \\ \sin 3(t+\frac{\pi}{2})=-\cos 3t $$ So : $$\lim_{t \to 0}\frac{1-\cos t}{\cos t-\cos…
Almot1960
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Find the constant and limit

Here is the limit $$\lim_{x\rightarrow-2} \frac{4x^2 + ax + a + 12}{x^2 + x - 2}.$$ a) Find the constant $a$ b) find the limit I dont think it is solvable because it didn't tell me as $x \rightarrow -2$, $y \rightarrow $?
Secret
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Limit $\lim_{x\to 0}\frac{\sin\left(\pi\cos^2(x)\right)}{x^2}$

Could someone help me solve this limit without L'Hopital's rule? $$ \lim_{x\to 0}\frac{\sin\left(\pi\cos^2(x)\right)}{x^2} $$
BuluBestTapu
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Sum rule in limits

Suppose we want to find $\lim_{x \to a}(f(x) - g(x))$ . We are not aware about the existence of $\lim_{x \to a} f(x)$ and $\lim_{x \to a} g(x)$ . Can we use sum rule and rewrite it to $\lim_{x \to a}f(x) - \lim_{x \to a}g(x)$ ? For example if after…
S.H.W
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fine the limits :$\lim_{x \to 0} \frac{(\sin 2x-2x\cos x)(\tan 6x+\tan(\frac{\pi}{3}-2x)-\tan(\frac{\pi}{3}+4x))}{x\sin x \tan x\sin 2x}=?$

fine the limits-without-lhopital rule and Taylor series : $$\lim_{x \to 0} \frac{(\sin 2x-2x\cos x)(\tan 6x+\tan(\frac{\pi}{3}-2x)-\tan(\frac{\pi}{3}+4x))}{x\sin x \tan x\sin 2x}=?$$ i know that : $$\lim_{x \to 0} \frac{\sin x}{x}=1=\lim_{x \to…
Almot1960
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Finding value of limit involving cosine

I know we can use Maclaurin expansion and l'hopital's rule for solving it but I want another way . Find value of $\lim_{x \to 0} \frac{\cos^2x - \sqrt{\cos x}}{x^2}$. My try : I multiplied and then divided by conjugate of numerator but it didn't…
S.H.W
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How do I calculate this limit $\lim\limits_{x \to 0}\frac{ 1-x^n}{1-x^m}$?

I'm having problem finding limit of lim $x \to 1$ for $(1-x^n)/(1-x^m$). I know that the result is $n/m$ but I can't really come up with how to modify the expression to arrive at that.
Zerg Overmind
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limit without L'Hôpitale rule or infinity series

I was solving this limit $$ \lim_{x\to0} \frac{\ln(x+1)-x}{1-\cos(x)}=L $$ I tried to rewrite the function $f(x)=\dfrac{\ln(xe^{-x}+e^{-x})}{2\sin^2(\frac{x}{2})}$ and we have $$\lim_{x\to0} f(-x)=L$$ $$ 4L=\lim_{x\to0}…
AbdulQader Qassab
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What is the following limit? $\lim_{x \to 0}xe^{\frac{1}{x}}$

I need to find lateral limits of this one $$\lim_{x \to 0}xe^{\frac{1}{x}}$$ I tried and I got that when $x$ is smaller than $0$ the limit is $0$. But what do I do when $x$ is bigger than $0$?
Ghost
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Find the limit without L'hopital rule $\lim_{ x \to 1}\frac{1-\cot(\frac{π}{4}x)}{\sin πx}=$?

Find the limit without L'hopital rule $$\lim_{ x \to 1}\frac{1-\cot(\frac{π}{4}x)}{\sin πx}=?$$ My Try: $$1-\cot( \frac{π}{4}x)=1-\frac{1}{\tan( \frac{π}{4}x)}=\frac{\tan( \frac{π}{4}x)-1}{\tan( \frac{π}{4}x)}\\\sin (πx)=\sin (\pi-\pi x)=-\sin…
Almot1960
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Limits without L'Hopital, what is the rationale for doing so

The fact that there is a tag for problems asking one to find a limit without use of L'Hopital's rule tells me this type of problems is not uncommmon. My question is: What is the rationale for 'find limit without L'Hopital's rule' problems? Namely,…
Jan
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Limit of a trigonometric function as it goes to infinity

What is the limit of $n {(\sin(x))^{2n+1}} \cos(x)$ as $n$ goes to infinity? The value of $\sin(x)$ and $\cos (x)$ are between $-1$ and $1$ so the limit is $\infty$?
Nitish
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