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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How to evaluate $\lim\limits_{x \to 0} \frac{\sqrt{1+x}-\sqrt{1-x}}{x}$?

I know it can be solved easily with L'H$\mathrm{\hat o}$pital's rule, but I am not allowed to use this rule. What I can use is the definition of function limit, rules of limit algebra (sum, product and quotient), and the composite rule. I am only…
IncredibleSimon
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Derivative of sinx from first principle

I am just confused over this step in the derivative of sin x $$ \lim_{x \to 0}{\sin\left(x\right) \over x} = 1 $$ When we have the limit, my textbook uses the small angle approximations, to say $\sin\left(x\right) \approx x$ for small $x$ in…
Nav Bhatthal
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Solving the limit $\lim_n \left(1+\frac{1}{-n} \right)^{-n}$

I was doing an exercise about limits of sequences and arrived at the following limit: $$\lim_n \left(1+\frac{1}{-n} \right)^{-n}\ \ \ \ (1)$$ We are supposed to solve the limit without using L'hopital's rule. The only limit that's similar to this…
Eduardo Magalhães
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Limit without L'Hopital rule - help!

I need help in the following problem: $$\lim_{x\to0} \frac{x - x \cos x}{\sin^2 2 x}.$$ L'Hopital has not yet given in my Calculus I class at college, so there has to be a way to solve it without L'Hopital, but I can't find it. My…
Thamine
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Limit of a function without LH

Evaluate the limit : ${((1+x)^{1/x} -e})/x$ when x tends to zero. This can be solved using L-Hopital rule , however I was wondering if there is any other method to do it, like any series formula or any other thing?
green_32
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Calculate the following limit without using L'Hôpital's rule

I need to calculate the following limit without using L'Hôpital's rule: $$ \lim_{x\to\infty} \big(\frac{x}{2}\big)^{\frac{1}{x-2}} $$ I written the expression using $ln$: $$ \lim_{x\to\infty} e^{\frac{ln(\frac{x}{2})}{x-2}} $$ I don't know how…
Daniel
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Is there a easy way to solve $\lim_{z \to 1} {{z^n -1} \over {z^m-1}}$ without L'Hospital?

Is there a easy way to solve $$\lim_{z \to 1} {{z^n -1} \over {z^m-1}}$$ without L'Hospital? Here $z \in \mathbb{C}$ and $m,n \in \mathbb{Z}$. Clearly $n/m $, using L'Hospital.
user77614
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Find the integer value of a, b such that $\lim_{x \rightarrow 4} \frac{ax-\sqrt{x}+b}{x-4}=\frac{3}{4}$ without using the L'Hôpital's rule.

Find the integer value of $a$, $b$ such that $\lim\limits_{x \rightarrow 4} \frac{ax-\sqrt{x}+b}{x-4}=\frac{3}{4}$ without using the L'Hôpital's rule. My work: I multiplied with $\frac{ax+\sqrt{x}+b}{ax+\sqrt{x}+b}$, and obtained: $=\lim\limits_{x…
Fiorella Susanto
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limits , need help

How would I go about finding the following limit: $$\lim_{n \to \infty} (2^n-n^2)^{\frac{1}{n}}$$
david david
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Limit of the form 0 $\cdot$ inf

I have been trying for hours to solve the following limit without using De L'Hospital theorem, but I got stuck. Can you help me? Thanks in advance! (The solution should be $-\sqrt2$) $\lim_{x\rightarrow\pi/4}(2\sin(x)-\sqrt2)\tan(x-3\pi/4)$
Simone
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How to progress: if $c \neq 0$, prove $\lim\limits_{n \to \infty} \frac{a \cdot n + b}{c \cdot n+d}$

So far have this; Need to show $\forall \epsilon >0 \; \exists N=N(\epsilon) \text{ s.t. } \forall n>N \; \Bigl{|}\frac{a \cdot n + b}{c \cdot n+d} - \frac{a}{c}\Bigr{|}<\epsilon \\$ So $|\frac{a \cdot n + b}{c \cdot n+d} - \frac{a}{c}| =…
alortimor
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Limit with tan and arctan $\lim_{x\to 2} \frac{\arctan x^2-\arctan 4}{\tan 2^x-\tan 4}$

Can someone help to do the following limit without L'Hospital or series expansion: $$\lim_{x\to 2} \frac{\arctan x^2-\arctan 4}{\tan 2^x-\tan 4}$$ With l'Hospital it is simple: $$\lim_{x\to 2} \frac{\arctan x^2-\arctan 4}{\tan 2^x-\tan 4}=\lim_{x\to…
user753925
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Evaluating $\lim_{x \to 0} \frac{x - \sin x \cos x}{\tan x - x} $ without L'Hospital or series expansion

Evaluate the limit without using L’Hospital’s rule and without using series expansion $$\lim_{x \to 0} \frac{x - \sin x \cos x}{\tan x - x} $$
cgo
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limit on infinity

I am trying to solve this question but I am getting a negative infinity which is wrong. $$\lim _{x\to \infty }\left(\ln(e^{2x}-1)-\frac{x^2-3}{x}\right)$$ $$=\lim _{x\to \infty }\left(\ln(e^{2x}-1)-\frac{x(x^{ }-\frac{3}{x})}{x}\right)$$ $$=\lim…
Feemo Fellow
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Solving indeterminate limit form without L'Hopital's rule

$$\lim_{x\to2}\frac{\sqrt[4]{x}-\sqrt[4]{2}}{\sqrt[3]{x}-\sqrt[3]{2}}$$ Using L'Hopital's rule the simplified form of the above limit is found to be $\frac{3}{4\sqrt[12]{2}}$ However, is it possible to simplify it without using derivatives?…
SolvingTrainee
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