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.

3046 questions
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Evaluate $\lim_{x \to 1}\frac{1-x^2}{\sin\pi x}$ without L'Hopital

I'm trying to evaluate the given limit without using L'hopitals rule. $$\lim_{x \to 1}\frac{1-x^2}{\sin(\pi x)}$$ Replacing $x$ by $1$ leads to $\frac{0}{0}$. I have tried multiplying by $\frac{1+x^2}{1+x^2}$ and resoving the $\sin x$ factor by…
TheAlPaca02
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Why $\lim\limits_{x \to 3} \frac{3-x}{\ln(4-x)}=1$?

I have $\lim\limits_{x \to 3} \frac{3-x}{\ln(4-x)}=1$ For $x \to 3$, I get: $\frac{0}{0}$ How to calculate it, without L'Hôpital's rule?
Dave
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Calculate limit $\lim_{x \to 2} \frac{(2^x)-4}{\sin(\pi x)}$ without L'Hopital's rule

How to calculate limit: $\lim_{x \to 2} \frac{(2^x)-4}{\sin(\pi x)}$ without L'Hopital's rule? If $x = 2$, I get uncertainty $\frac{0}{0}$
Dave
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Find $\lim_{x \to 0} \frac{e^{2x}-1}{\tan x}$

I'd like help finding $$\lim_{x \to 0} \frac{e^{2x}-1}{\tan x}$$ without the use of L'Hôpital's rule. So far I did this: $$\lim_{x \to 0} \frac{e^{2x}-1}{\tan x}$$ $$=\lim_{x \to 0} \frac{\cos x(e^{2x}-1)}{\sin x}$$ $$=\lim_{x \to 0} \frac{x\cos…
Jack Pan
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Limit of $a\sqrt{x+1}+b\sqrt{4x+1}+c\sqrt{9x+1}$ when $x\to\infty$, for given real numbers $a$, $b$, $c$

Find the limit of $a\sqrt{x+1}+b\sqrt{4x+1}+c\sqrt{9x+1}$ when $x\to\infty$, for given real numbers $a$, $b$, $c$. I would like to see a solving method without l'Hopital or Taylor expansion.
Ghost
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How to solve these limits using a formula for logarithm limit(without applying L'Hopitale rule)

How to solve these limits using a formula for logarithm limit(without applying L'Hopitale rule) $$ \lim_{x \to 0} \frac{\sqrt{1 + \tan(x)} - \sqrt{1 + \sin(x)}}{x^3} $$ $$ \lim_{x \to 0} \frac{\arctan 2x}{\sin[2 \pi(x+10)]}$$ I suppose in the second…
PeterEliot
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another trig limit without L'Hospital?

$$\lim_{x\to \pi/3 }\dfrac{1-2\cos\left(x\right)}{{\pi}-3x}$$ Here's what I tried: ${\pi}-3x=y$, $\dfrac{{\pi}-y}{3}=x$ $$\lim_{y\to\ 0} \dfrac{1-2\cos\left(\frac{{\pi}-y}{3}\right)}{y}$$ ...
lohe
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How to solve this limit without (L'Hôpital's rule)

I need to solve this limit without L'Hôpital's rule. These questions always seem to have some algebraic trick which I just can't see this time. $$ \bbox[yellow] { \lim_{x\to 2} \left( \frac{e^2-e^x} {2-x} \right) } $$ Could someone give me a hint…
AbdulQader Qassab
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Question of limit 1/log(x without lhopital

$\lim_{x\to\infty}{{\log \log \left(1+{{1}\over{x}}\right)}\over{\log x}}$ I cant do it. If somebody can do the question for me. I need do to the test but I not allowed use l'hopital.
user340886
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How to compute $\lim\limits_{x\to \frac{\pi}{2}}\frac{\sin(x)-1}{x-\frac{\pi}{2}}$ without using the L'Hôpital's rule?

$$\lim\limits_{x\to \frac{\pi}{2}}\frac{\sin(x)-1}{x-\frac{\pi}{2}}$$ I tried to compare this function with the derivate definition formula $$\lim\limits_{x\to a} \frac{f(x)-f(a)}{x-a}$$ And I did find the correct solution, but I'm not sure that…
Isabel
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Prove that $\lim_{x\to 0} \frac{a^x-1}{x}=\ln a$

Without using L'Hopital's rule, prove that $$\lim_{x\to 0} \frac{a^x-1}{x}=\ln a$$
Mohamed Mostafa
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Solving $\lim\limits_{x \to 0}\frac{\ln \cos 3x}{\ln \cos (-x)}$ without L'Hospital rule

How can I solve this limit not using L'Hospital rule? $\lim\limits_{x \to 0}\frac{\ln \cos 3x}{\ln \cos (-x)}$ Thank you very much.
heky__
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Calculating $\lim_{x\to\infty} (\sin\frac{1}{x}+\cos\frac{1}{x})^x$ without l'Hopital

Calculate $\lim_{x\to\infty} (\sin\frac{1}{x}+\cos\frac{1}{x})^x$ without using l'Hopital's rule. I attempted pulling something out to get a limit that resembles $e$, this gave…
Nescio
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Limit solving without using L'Hopital rule

How can I solve this limit $$ \lim_{x \to 0}\frac{e^{x+1}-e}{3x} $$ without using L'Hopital's rule? I know this is true: $$ \lim_{x \to 0}\frac{e^{x}-1}{x} = 1 $$ So i belive we have to use this in anyway possible.
João Silva
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Limit of a square root with $x^2+x$

I saw the following problem: $$\lim_{x\to \infty} \sqrt{9x^2+x}-3x$$ My first thought was to say that the $x$ term is overpowered when $x$ becomes large enough, so the square root becomes just $\sqrt{9x^2} = 3x$, and the value of the limit is…
Ypnypn
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