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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Calculating limit in the infinity

Consider we want to find $\lim_{x \to -\infty} x(x+\sqrt{x^2 - 8})$ . I know the answer is this: But in the last step $\sqrt{x^2-8}$ has been changed to $\sqrt{x^2}$. I know we are dealing with infinity and $\infty - 8$ is also infinity but I'm…
S.H.W
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Limit going to infity for $(x + (x +(x)^{\frac 12})^{\frac 12})^{\frac 12} / \sqrt{x+1}$

This one is giving me some truble to modify correctly to allow me get $$\lim\limits_{x\to+\infty} \frac{(x + (x +(x)^{\frac 12})^{\frac 12})^{\frac 12}}{(x+1)^{\frac 12}} = 1$$ I keep on trying fatoring and turning it around but i can't seem to get…
Zerg Overmind
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What is the value of the limit?

I need to solve the limit $$\lim_{x\to0}\frac{e^{-\frac{1}{x^2}}}{x}$$ If I replace $x$ with $0$ I get $\frac00$ so that is not what I should do. Do you have any idea how to do it?
Ghost
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$\lim_{x\to 0}\frac{\sin^2x-x\tan x}{x^4}$

fine the limit : $$\lim_{x\to 0}\frac{\sin^2x-x\tan x}{x^4}$$ My Try : $$\lim_{x\to 0}\frac{\sin^2x-x\tan x}{x^4}\\\lim_{x\to 0}\frac{\frac{\sin x \sin x}{x}-\frac{x\tan x}{x}}{x^4/x}\\\lim_{x\to 0}\frac{\sin x -\tan x}{x^3}=-1/2 $$ is it right ?
user402750
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$\lim_{ x \to4 }\frac{x\sqrt{x}+2x-5\sqrt{x}-6}{x\sqrt{x}-8}=?$

fine the limits : $$\lim_{ x \to4 }\frac{x\sqrt{x}+2x-5\sqrt{x}-6}{x\sqrt{x}-8}=?$$ my try : $$\lim_{ x \to4 }\frac{x\sqrt{x}+2x-5\sqrt{x}-6}{x\sqrt{x}-8}=HOP\\\lim_{ x \to4 }\frac{\dfrac{3x+4\sqrt{x}-5}{2\sqrt{x}} }{\dfrac{3\sqrt{x}}{2} }=5/4$$ BUT…
Almot1960
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$\lim_{x\to 1/2} \frac{\cos(\pi x)}{4x^2-1}$

find the limit : $$\lim_{x\to 1/2} \frac{\cos(\pi x)}{4x^2-1}$$ my try : $$\lim_{x\to 1/2} \frac{\cos(\pi x)}{4x^2-1}=\lim_{x \to 1/2}\frac{\cos (\pi x-\pi)}{\pi x-\pi} \times \frac{\pi (x-1)}{4x^2-1}\\=\lim_{x\to 1/2} \frac{\cos(\pi…
Almot1960
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Calculate limit $\lim_{x \to 2 \pm 0} \frac{(x+1)^2}{2-x}$ without L'Hopital's rule

I have $\lim\limits_{x \to 2 \pm 0} \frac{(x+1)^2}{2-x}$ I calculated $\lim\limits_{x \to 2 + 0} \frac{(x+1)^2}{2-x}=\lim\limits_{x \to 2 - 0} \frac{(x+1)^2}{2-x}=\lim\limits_{x \to 2 } \frac{(x+1)^2}{2-x}=\frac90=\infty$ Or is it wrong?
Dave
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How to evaluate $\lim_{x \to1}\frac{x- \ln x-1}{(x-1)^2}$ without L'Hopital?

I will be thankful if somebody help me to solve this limit without L'Hospital's rule $$\lim \limits_{x \to1}\frac{x- \ln x-1}{(x-1)^2}$$
Lind
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Solving this limit without L'Hopital

I'm trying to find the solution for the following limit without using L'Hopitals rule. The indeterminate form of $\frac{0}{0}$ is obtained but both the conjugate and or squeeze theorem can't be applied here (I think). I know that the solution is…
TheAlPaca02
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Find this limit without using L'Hôpitale

Can any give me a hint to solve this limit $ \lim_{x\to 1} \frac{1}{x-1}-\frac{1}{ln(x)} $ I used L'Hôpitale rule and I found that the solution is -$\frac{1}{2}$ And I tried another ways like $ T=x-1$ $\lim_{t\to 0}…
AbdulQader Qassab
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How do I derive $\lim_{x\to\infty}(1+1/x)^x,\quad x\in\mathbb{R}$ from $\lim_{n\to\infty}(1+1/n)^n=e,\quad n\in\mathbb{N}$?

How do I derive $$\lim_{x\to\infty}(1+1/x)^x,\quad x\in\mathbb{R}\tag{1}$$ from $$\lim_{n\to\infty}(1+1/n)^n=e,\quad n\in\mathbb{N}\tag{2}$$ ? Note: Without calculus (continuity, derivative, aso.). (2) is only an idea, every method using…
PeptideChain
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Limit without using the L'HÔPITAL rule:$\lim_\limits {x \to \pi} \frac{(e^{\sin x} -1)}{(x-\pi)}$

$$\lim_\limits {x \to \pi} \frac{(e^{\sin x} -1)}{(x-\pi)}$$ I found $-1$ as the answer and what I did was: $\lim_\limits {x \to \pi} \frac{(e^{\sin x} -1)}{(x-\pi)}$ $\Rightarrow$ $\lim \frac{(f(x) - f(a))}{(x-a)}$ $\Rightarrow$ $f(x)=(e^{\sin…
user331011
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Finding the limit inf./inf. form

Find the limit of the sequence defined by Lim $_{n \to \infty} \frac{(2n)!}{(2^n\cdot n!)^2}$ How to start? Is there another way of showing this sequence is convergent without finding the limit?
rahul kumar
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Prove this limit - exponentials

$$\lim_{x\rightarrow 0} \frac{4^{-1/x}}{3^{-1/x}+5^{-1/x}}$$ L'Hôpital doesn't work... I know the limit is zero.
user118443
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limit of a function being a sum and product of trignometric and hyperbolic functions

I need to know the $$\lim_{(x,y)\to(0,0)}{\sin x-\sinh x\over 2x^2+y^2-xy}\ln(x^2+y^2)$$I used finite expansion on the hyperbolic and trigonometric function and then used polar coordinates where the limit tends then to $0.$ Am I right ?
mandez
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