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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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Do we need to apply the L'Hôpital's rule to all the polynomial when evaluating infinity?

I have this: $$-\frac{x}{e^{\frac{x}{10}}}-\frac{10}{e^{\frac{x}{10}}}$$ Do I need to derivate the numerator and denominator of all this expression or just : $$-\frac{x}{e^{\frac{x}{10}}}$$ And $$-\frac{10}{e^{\frac{x}{10}}}$$ Keeps the…
Limanido
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Limit without Lhospital rules

please how can i solve this limit ((1/cos x)-(1/cos a))/(x-a) where x aproches a. Thank you for you help. Unfortunately i dont understand the way how to solve limits of this type.
Robin Pastrňák
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What to do when L'Hopital's rule does not work

I am looking to calculate the value of the function $\frac{(ax+1)^{-(1+\frac{1}{ax})}\ln(ax+1)}{x}$ when $x \rightarrow 0$, and $a$ is a positive constant. Repeated application of L'Hopital rule keeps giving an indeterminate form. Any suggestion…
King008
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$\lim_{x \to 0} \frac{\cos(\sqrt{a+x})-\cos(\sqrt{a})}{x}$

$$\lim_{x \to 0} \frac{\cos(\sqrt{a+x})-\cos(\sqrt{a})}{x}$$ The answer to this question is $\frac{d}{da}\cos(\sqrt{a})$ i.e., $\begin{eqnarray*}-\frac{\sin(\sqrt{a})}{2\sqrt{a}}\end{eqnarray*}$ but I want to get answer by using general rules of…
Sunil kumar
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How to solve this limit without the L'Hospital?

How would one evaluate the following limit without using L'Hospital Rule $$\lim_{x\to -1}\dfrac{\sin(x+1)}{x^3+1}$$ the result should be $1/3$.
qua
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I need to solve this limit without using L'Hospital's rule

I need to solve this limit without using L'Hospital's rule. $$\lim_{x\to 0}⁡(\cos x)^\left(\frac{-4}{x^2}\right)⁡$$
Alexandra
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Hints to solve $\lim_{x \to 0^+}{\frac{\sqrt x +\tan^3x+\sqrt x\sin^2x}{x+x^2\cos x-\tan^2x}}$ without L'Hopital

$$\lim_{x \to 0^+}{\frac{\sqrt x +\tan^3x+\sqrt x\sin^2x}{x+x^2\cos x-\tan^2x}}.$$ I tried divide both terms by $x^2$ but I didn't get anywhere doing this. Can someone give me some hints on how to solve this limit?
Levio
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Evaluating the limit of $x^2 (1-\cos\frac{1}{x})$ when $x$ approaches infinity

I wanted to evaluate the limit $$\lim_{x\to\infty}x^2(1-\cos\frac{1}{x})$$ Since we know that $-1\leq \cos x\leq1$ and that $-1\leq \cos\frac{1}{x} \leq 1$, so by algebraic manipulation, $0\leq x^2(1-\cos\frac{1}{x})\leq2x^2 $. Why does squeeze…
Loo Soo Yong
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Solving limit $ \lim_{x\rightarrow 1}\left(\frac{\sqrt[3]{7+x^3}-\sqrt[2]{3+x^2}}{x-1}\right) $ without L'Hopital.

I was trying all the day to resolve this problem with different method without L'Hopital but I can't do it, I would really like to up my mathematical development but the post doesn't allow me because I have less than 10 of reputation to up a image,…
mrpepo877
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Is it possible to solve $\lim_{x\to a}\frac{\sin x - \sin a }{x-a}$ without derivatives?

My teacher replaced $x-a = t$ and then said as $x$ approaches $a$ we have $a-a=t$ so $t$ approaches $0$ and then said lim as $t$ approaches $0$, $\frac{\sin(t+a)-\sin a}t = \lim_{t\to0}\frac{\sin t\cos a + \sin a\cos t - \sin a}t$ and then she…
Endrit Shabani
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Limit of $\lim_{x\to0} \left(\frac{1}{x}-\cot x\right) $ without L'Hopital's rule

Is it possible to evaluate $$\lim_{x\to 0} \left(\frac{1}{x}-\cot x\right) $$ without L'Hopital's rule? The most straightforward way is to use $\sin{x}$ and $\cos{x}$ and apply the rule, but I stuck when I arrived to this part (since I don't want…
Loo Soo Yong
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Limits with Trig

Is there any way to evaluate $$\lim_{x\to0} \frac{x \cos x - \sin x}{x^2}$$ without using L'Hopital's Rule? I was trying to use some of the standard trig limits (e.g. $\lim_{x\to 0} \frac{\sin x}{x} = 1)$, etc. but couldn't figure it out. Thank you.
Conan Wong
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$ \lim_{n \to \infty} \frac{\log n^3}{\log (n^3+3n^2)}$

I want to solve this $$ \lim_{n \to \infty} \frac{\log n^3}{\log \left(n^3+3n^2\right)}$$ I found that $$ \frac{\log n^3}{\log \left(n^3+3n^2\right)}<\frac{\log\left( n^3\right)}{\log \left(n^3\right)}=1$$ but I need another bond according to the…
Anne
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Proving infinite limit $\frac{polynomial}{exponential}$, only using definition of limit

I have problems finding a proof for limit of the sequence: $(a_n) = \frac{n^4}{3^n}$ as $n \rightarrow \infty$ or $\lim\limits_{x \to \infty} \frac{n^4}{3^n} = a^*$ Exponential functions grow faster than polynomials, so I know that $a^*$ should be…
FullMetal Pancake
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I need help solving a limit, without using l'hôpital's rule.

The limit is $$\lim_{x\to0}\left(\frac{\ln(1+x)-\ln(1-x)}{x}\right)$$ Please resolve without using l'hôpital's rule, haven't made it that far yet. Thanks in advance.
Alvaro Morales
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