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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Finding the limit of $\lim_{x\rightarrow 0}\frac{\sqrt[m]{1+P(x)}-1}{x}$

I need to find the limit of $\lim_{x\rightarrow 0}\frac{\sqrt[m]{1+P(x)}-1}{x}$ where $P(x) = a_{1}x + a_{2}x^2+...+a_{n}x^n = \sum_{i=1}^{n}a_{i}x^{i}$ My try. let $n=\frac{1}{m}$ $\lim_{x\rightarrow 0}…
shcolf
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Finding the limit of a trigonometric function

I got it interms of e but couldn't find the limit of the power $$\displaystyle \lim_{x \to \frac{\pi}{4}} {\sin 2x}^{\tan^2 2x}$$
E and pi
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Minimizing denominator when calculating limit of function

I calculated a limit of function as follows: $$ \begin{array}{ll} \lim_{x \to 1}\frac{x - 1}{x^2 - 1} = \\ \\ \quad = \lim_{x \to 1}\frac{x - 1}{(x + 1)(x - 1)} = \\ \\ \quad = \lim_{x \to 1}\frac{1}{(x + 1)} = \\ \\ \quad =…
Dor
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Solve $\lim _{x\to 0} \frac{e^x-\ln\left(1+x\right)-1}{x^2}$ without L’Hospital’s rule

I was making some limits using Maclaurin’s series and one of those was: $$\lim _{x\to 0} \frac{e^x-\ln\left(1+x\right)-1}{x^2}$$ While this limit can be solved using Maclaurin’s series or L’Hospital’s rule, I want to find out a solution without…
Neutron
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Limit of $\frac{\log(e+x)-1}{x}$ as $x\to 0$ without using L'Hopital

I was doing a derivative by definition and I need to solve that limit using equivalent infinitesimals and such, without L'Hopital Rule. Any hint?
user10000024
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How to solve $\lim_{x\to\infty} \left(\frac{4+x}{1+x}\right)^{4+2x}$ (without l'Hospital maybe?)

How to solve $$\lim_{x\to\infty} \left(\frac{4+x}{1+x}\right)^{4+2x}?$$ Well, it looks similar to: $$e = \lim_{n\to\infty}\left( 1 + n^{-1}\right)^n,$$ but I can't get it solved. Sure, I could use l'Hospital twice (as some calculators suggest). But…
Korrola
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2 answers

Where am going wrong in finding this limit?

I have been asked to find the following limit $f(x)=(x^{2n}-1)/(x^{2n}+1)$ as n tends to infinity. The answers given are in MCQ type with more than one answer correct. I'll just write the correct answers. a)$f(x)=1$,for$|x|>1$ and b) $f(x)=-1$, for…
Natasha J
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Calculate $\lim_{x \to a} \frac{\log(x-a)}{\log(e^x-e^a)}$ without using L'Hopital

I can calculate this easily using L'Hopital Rule. Can anyone give me some pointers on how to do this without using L'Hopital? $$\lim_{x \to a} \frac{\log(x-a)}{\log(e^x-e^a)}$$ I tried substitution by $n = x+a$ and then $\lim_{n \to 0}$ but that…
sato
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Find limit (without using L' Hospital Rule) I can find this limit using L' Hospital Rule, I do not know how to do it without that

Find limit (don't use Lophital rule) $$\lim _{x\to 0}\left(\frac{\sqrt{1+x}\:-\sqrt{1-x}}{\sqrt[3]{1+x}-\sqrt[3]{1-x}\:}\right)$$ I can find this limit using L' Hospital Rule, I do not know how to do it without that.
Naz a
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Limit of $\lim_{x\to1} \frac{\sin{\ln {x}}}{x^5-7x^3+6}$

How does one evaluate this limit without using L'hopital's rule? $$\lim_{x\to1} \frac{\sin{\ln {x}}}{x^5-7x^3+6}$$ I tried to use the substitution $u=\ln x$ but all I get is an exponential polynomial which can be factored, but it doesn't seem to…
Loo Soo Yong
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4 answers

prove $\lim\limits_{x \to \infty}(1+ 1/n)^n = e$ without using l'Hospitals rule

I did $y=(1+1/n)^n)$ so $ln(y)=n*ln(1+1/n)$ so then $ln(y)=Ln(1+1/n)/(1/n)$ which sith l'hospitals rule becomes $ln(y)=(-1/(n^2 + n))/(-1/n^2)$ which simplifies to $n/(n+1)$ which using l'hospitals rule is $1$ so the answer is e. how do you solve…
Ekadh Singh - Reinstate Monica
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How can I find a limit using equivalent functions and substitutions ( without applying L'Hospital) for the following problem:

How can I find a limit using equivalent functions and substitutions ( without applying L'Hospital) for the following problem? There you can see the expression, I need to find the limit for, as x tends to 0 $\exp[(\cos(\sqrt x)-1)/x]$, square…
user
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2 answers

Trouble with a sequence limit

I'm having problems with this sequence limit. Note: I am not allowed to use L'Hopital's rule. $$\lim_{n\to\infty} \frac{\sqrt{n^2+1}-n}{\sqrt{n^3+1} - n\sqrt{n}}$$ Thanks in advance ^^.
Boxonix
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find the limit of the following function without using L'Hopital's Rule

Can help to solve the following limit without using L'Hopital's Rule? $$\lim_{x\to 0}\frac{e^x-x-1}{cosx-1}$$
qsmy
  • 545
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3 answers

What is $\lim_{x\to\infty} \frac{\sqrt[4]{x^5} + \sqrt[5]{x^3} + \sqrt[6]{x^8}}{\sqrt[3]{x^4 + 2}}$?

How would you calculate this limit without use of derivatives? I know it goes to 1 but i can't seem to arrive at it. $$\lim_{x\to\infty} \frac{\sqrt[\large4]{x^5} + \sqrt[\large 5]{x^3} + \sqrt[\large6]{x^8}}{\sqrt[\large 3]{x^4 + 2}}$$
Zerg Overmind
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