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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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What is the limit of this function as x tends to 0

I am trying to evaluate the limit of $x[1/x]$ as x tends to zero, where $[.]$ is greatest integer function. I know this is dumb question but can I write [1/x] as 1/[x]? Thanks in advance.
Natasha J
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Two limits without L’Hopital

$$\lim_{x \to \infty}\frac{\ln(x^3-5x+3)}{\ln(x^5-6x^2-6)}$$ $$\lim_{x\to \infty}\frac{\ln(1+e^{3x})}{x}$$ For the first function I intended using $\lim_{x\to \infty}\frac{\ln x}{x}=0$ but I couldn’t figure it out. For the second limit I have no…
furfur
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Limit at infinity of $\frac{\ln x}{x-1}$

I don't know how to prove that $\lim_{x\to\infty}\frac{\ln x}{x-1}=0$ without using L'Hopital. I've tried to use the definition of $\ln x=\int_{1}^{x}\frac{1}{t}dt$ and the fact that $\frac{x-1}{x}<\ln x
Iesus Dave Sanz
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Limit without de l'Hôpital: $\lim _{x\to 0\color{red}{\boldsymbol -}}\left(1+x^3\right)^{\frac{1}{\left(x^2+1\right)^4-1}}$

I have this limit of this form $$f(x)^{g(x)}=e^{g(x)\ln(f(x))}$$ $$\lim _{x\to 0\color{red}{\boldsymbol -}}\left(1+x^3\right)^{1/\left((x^2+1)^4-1\right)}$$ In our case I can write in the…
Sebastiano
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Using definition of limit to prove that $\lim\limits_{x \to 1}\frac{2-x}{4-x}=\frac{1}{3}$

I hit a block when discovering a negative $\delta$. This is how: I need to show that$$\forall \epsilon>0 \; \exists \delta>0 \text{ s.t. } \mid x-1 \mid < \delta \Rightarrow \Bigl| \frac{2-x}{4-x}-\frac{1}{3} \Bigr| < \epsilon$$ To find such a…
alortimor
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Limit of $\lim_{x\to 3} \frac{3-x}{6\sin \frac{\pi x}{18}-x}$

I want to solve a limit without l'Hospital, just with algebraic manipulation: $$\lim_{x\to 3} \frac{3-\sqrt{6+x}}{6\sin \frac{\pi x}{18}-x}$$ I started with: $$\lim_{x\to 3} \frac{3-\sqrt{6+x}}{6\sin \frac{\pi x}{18}-x}=\lim_{x\to 3}…
user754302
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Calculate limite: $\lim_{x \to 2} \frac{\cos(\pi/x)}{2-x}$ without L'Hospital's rule.

I need to calculate the following limit without using L'Hospital's rule: $$\lim_{x \to 2} \frac{\cos(\pi/x)}{2-x}$$
Cironis
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How to solve this limit without the L'hopital's rule

While practicing for my high school calculus exam, I went through the following limit problem: $$\lim_{x \to 0}x\cdot \sqrt{\cos{\frac{1}{x}}}$$ We haven't covered any similar example, and although we learned about the L'hopital's rule, our teacher…
someone123123
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How to determine a limit without L'Hospital's rule

How to solve this type of limit without L'Hospital rule. $$\lim_{x\to a}\frac{a^x-x^a}{x-a}$$
Robin Pastrňák
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Solving limit $\lim_{x\to0^+}\cos(\sqrt{x})^{1/x}$ without l'Hospital's rule

How to solve this limit $$\lim_{x\to0^+}\cos(\sqrt{x})^{1/x}$$ without L'Hospital's rule.
Robin Pastrňák
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Find the two limits without the use of l'Hospital's rule or series expansion.

I was asked to evaluate these two limits: $$\lim_{x\rightarrow0}\frac{x^3}{x-\sin x}$$ $$\lim_{x\rightarrow0}\frac{e^{-x^2}+x^2-1}{\sin(3x^4)}$$ For the first one I tried to divide the numerator and denominator by $x^3$, but I can't get the answer…
Loo Soo Yong
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Generalization of $ \lim_{x\rightarrow\infty}\left(\sqrt[n]{a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0}-\sqrt[n]{a_n}x\right) $

Yes, I read https://math.stackexchange.com/a/3096211/578535 Then it made me think about the generalization of the following. $$ \lim_{x\rightarrow\infty}\left(\sqrt[n]{a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0}-\sqrt[n]{a_n}x\right) $$ From my problem…
Pizzaroot
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Definitionally prove that $\lim_{x \to 0}\frac{f(x)-f(0)}{x^2} = \frac{f''(0)}{2}$

$$\lim_{x \to 0}\frac{f(x)-f(0)}{x^2} = \frac{f''(0)}{2}\quad (f'(0) = 0)$$ It seems quite a rudimentary problem, but I can't find an appropriate solution without using L'hospital's rule and Maclaurin series. Is it possible that a problem can not…
Riddle Aaron
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limit of $\sqrt{(x^2+1)/(x^3+1)}$ as $x$ approaches negative infinity

$\lim_{x \rightarrow - \infty } \sqrt{ \frac{x^2+1}{x^3+1} } $ My teacher says that no limit exists, but Wolfram Alpha says the limit is 0. I'm confused. Any helps are welcome.
Tom Le
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$\lim_{x \to 2} \frac{\cos{\left(\frac{\pi}{x}\right)}}{x-2}$ without using De L'Hospital

$$\lim_{x \to 2} \frac{\cos{\left(\frac{\pi}{x}\right)}}{x-2}$$ This limit is supposed be found without L'Hospital's Rule, but I have not been able to get close to the answer using conjugates, squares, Pythagorean Identity, half angle formulas or…
hatinacat2000
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