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.

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Solve $ \lim _{x\to 0}\left(\frac{\sin x}{\sqrt{1-\cos x}}\right) $

I have to solve $$ \lim _{x\to 0}\left(\frac{\sin x}{\sqrt{1-\cos x}}\right) $$ How many times do I have to apply L'Hopital in this?
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Limit $\lim_{x\to \frac{\pi}{3}} \frac{\sin(x-\frac{\pi}{3})}{1-2 \cos{x}}$

I need to evaluate following limit: $$\lim_{x\to \frac{\pi}{3}} \frac{\sin(x-\frac{\pi}{3})}{1-2 \cos{x}}$$ Tried multiplying with the argument inside the sinus function but finished with this limit: $$\lim_{x\to \frac{\pi}{3}} \frac{x -…
Dan We
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Limit without l'hopital $\lim_{x \to 0} \left ( \frac{1}{x} -\frac{1}{e^x-1}\right )$

I want to find the limit without L'hopital's rule and without Taylor series: $ \lim_{x \to 0} \left ( \frac{1}{x} -\frac{1}{e^x-1}\right ) $ Is it possible? Hints are more than welcome! Thanks!
Lilly
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A complicated limit involving floor function

Let $f(x) = \lfloor x\lfloor1/x\rfloor \rfloor $ . Find $\lim_{x \to 0^{+} } f(x) $ and $\lim_{x \to 0^{-} } f(x)$ . I think $\lim_{x \to 0^{+} } f(x)$ doesn't exist but I have no idea about $\lim_{x \to 0^{-} } f(x)$ .
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Limit of $h(x)$ as $x \to \infty$

Let $h(x) = \sqrt{x + \sqrt{x}} - \sqrt{x}$ , find $\lim_{x \to \infty } h(x)$ . I've tried substitution , multiplying by conjugate and l'hospital's rule but didn't work .
S.H.W
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Limit of a function $\mathbb{R}^3 \to \mathbb{R}$

Limit of the function $$\lim\limits_{x,y,z\to 0,0,0} \frac{1}{xyz}\tan\bigg(\frac{xyz}{1+xyz}\bigg)$$ If this was a one-dimensional function, this would look like an oportunity to apply the limit $\frac{\sin(x)}{x} = 1$. Is there a way to substitute…
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Limit of $\frac{x^5-1}{x^2-1}$

I need to determine if the following limit exists. $$\lim_{x\to 1}\frac{x^5-1}{x^2-1}$$ I've already proved using L'Hospital that this limit exists and should equal to $\frac{5}{2}$, but unfortunately I'm not allowed to used anything more than basic…
blub
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Continuity of trigonometric function with absolute values

I have to evaluate the continuity of this function $$ f(x) = \begin{cases} \sin x + \dfrac{\sqrt{1-\cos2x}}{\sin x}, & x \neq 0 \\[6px] \sqrt{2}, & x=0 \end{cases} $$ For the function to be continuous we need $$ \lim_{x \to 0}f(x) = f(0) $$ We…
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How to compute this multi-variable limit?

I'm trying to review for my final and need help solving this problem that was given in a past homework assignment. The answer I keep getting is $-4/3$, but the correct answer is the limit DNE. I know I can approach this problem by testing different…
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Limit of $ \frac1{n -\log n}$ as $n$ approaches $\infty$.

I am not able to find the following limit. $$\lim_{n\to \infty} \frac{1}{n-\log n}$$ I tried replacing log function with it's expansion but soon stuck. Also tried dividing both numerator & denominator by $n$ to get the following $$\lim_{n\to \infty}…
Anuj
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Solving $\lim_{x\to 0}\frac{\ln(1+x)}x$ without De L'Hospital

Trying to solve this limit without derivatives I found this answer that is pretty straightforward and I can easily follow the flow. I can understand why ${u\to \infty}$ because: $$\lim_{u\to\infty}(1 + \frac{1}u)^u = e $$ but how there is a relation…
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Limit using Riemann integral

Okay so I have to find the following limit of the function given as : $$\lim_{n\to\infty} \left(\frac{1}{n}\cdot\frac{2}{n}\cdot\frac{3}{n}\cdots\cdots\frac{n}{n}\right)^\frac{1}{n} $$ Now , on taking $log$ both sides and rearranging , I get…
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Find a limit of a function W/OUT l'Hopital's rule.

I've got an expression: $\lim_{x\to 0}$ $\frac {log(6-\frac 5{cosx})}{\sin^2 x}$ The question is: how to find limit without l'Hopital's rule?
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Limit points of $z^n$ ($z\in\mathbb{C}$)?

The exercise is to find the limit points of $z^n$ where $z\in\mathbb{C}$ is a complex number? However, if $z=-1$ we have the limit points $1$ and $-1$, for $z=1$ we have the limit point $1$ and for $z=2$ we have no limit point. So how can you…
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Limit $\lim\limits_{(x,y)\to (0,0)}{\sin(xy)-xy\over x^2y}$

I need to use Taylor's theorem to compute: $\lim\limits_{(x,y)\to (0,0)}{\sin(xy)-xy\over x^2y}$ Using the theorem we have that: $\sin(xy)=xy-{(xy)^3\over6}+R_6(x,y)$ where $\lim\limits_{(x,y)\to (0,0)}{R_6(x,y)\over (x^2+y^2)^3}=0$ So…
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