Linked Questions

3
votes
3answers
107 views

Evaluating $\lim_{x\to0}\frac{x\sin{(\sin{x})}-\sin^2{x}}{x^6}$ [duplicate]

Evaluate: $$\lim_{x\to0}\frac{x\sin{(\sin{x})}-\sin^2{x}}{x^6}$$ I have been trying to solve this for $15$ minutes but sin(sin(x)) part has me stuck. My attempt:...
1
vote
1answer
32 views

Solve the following limit as $\lim_{x \to 0}$ [duplicate]

$$\lim_{x \to 0} \frac{x\sin(\sin x) - \sin^2 x}{x^6}$$ **My Attempt: ** I started with L'Hopital's rule. But it quickly became messy. So, I did not continue. I tried to write the Taylor series of ...
95
votes
3answers
5k views

Are all limits solvable without L'Hôpital Rule or Series Expansion

Is it always possible to find the limit of a function without using L'Hôpital Rule or Series Expansion? For example, $$\lim_{x\to0}\frac{\tan x-x}{x^3}$$ $$\lim_{x\to0}\frac{\sin x-x}{x^3}$$ $$\...
7
votes
4answers
7k views

$\lim_{x\to0} \frac{x-\sin x}{x-\tan x}$ without using L'Hopital

$$\lim_{x\to0} \frac{x-\sin x}{x-\tan x}=?$$ I tried using $\lim_{x\to0} \frac{\sin x}{x}=1$. But it doesn't work :/
5
votes
1answer
497 views

Is it possible to solve this limit without Hopital / Taylor / derivatives?

It's simple to prove with Hopital that $$ \lim_{x \to 0} \frac{x-\sin(x)}{x^3} = \frac{1}{6}$$ Is it possible to solve this limit without Hopital or Taylor (without derivatives)?
2
votes
3answers
3k views

limit $\lim_{x\to 0}\frac{\tan x-x}{x^2\tan x}$ without Hospital

Is it possible to find $$\lim_{x\to 0}\frac{\tan x-x}{x^2\tan x}$$ without l'Hospital's rule? I have $\lim_{x\to 0}\frac{\tan x}{x}=1$ proved without H. but it doesn't help me with this complicated ...
2
votes
5answers
178 views

Limit of a trig function. (Without using L'Hopital) [duplicate]

I'm having trouble figuring out what to do here, I'm supposed to find this limit: $$\lim_{x\rightarrow0} \frac{x\cos(x)-\sin(x)}{x^3}$$ But I don't know where to start, any hint would be appreciated,...
1
vote
4answers
210 views

How to Find the Minimum Value for $\frac{\tan x}{x}$ (piecewise defined)?

$$F (x) = \begin{cases} \displaystyle \frac{\tan(x)}{x} & x \not=0 \\ 1 & x=0 \end{cases} $$ How do I prove that there is a local minimum at $x=0$?
3
votes
3answers
153 views

Calculating limit of function

To find limit of $\lim_{x\to 0}\frac {\cos(\sin x) - \cos x}{x^4} $. I differentiated it using L Hospital's rule. I got $$\frac{-\sin(\sin x)\cos x + \sin x}{4x^3}\text{.}$$ I divided and multiplied ...
1
vote
5answers
410 views

Error calculating the limit $ \lim_{x \to 0}{\frac{x-\tan(x)}{x^2 \cdot \sin(x)}} $

Given this limit: $$ \lim_{x \to 0}{\frac{x-\tan(x)}{x^2 \cdot \sin(x)}} $$ Wolfram says the result is $\frac{1}{3}$ , but I tried to solve it and I get 0: $$ \lim_{x \to 0}{\frac{x \cdot (1-\...
2
votes
5answers
150 views

How to calculate $\lim_{x\to 0}\left(\frac{1}{x^2} - \frac{1}{\sin^2 x}\right)^{-1}$? [closed]

$$f (x) = \frac{1}{x^2} - \frac{1}{\sin^2 x}$$ Find limit of $\dfrac1{f(x)}$ as $x\to0$.
3
votes
5answers
148 views

How to compute $\lim_{x\to 0^+}\frac{\arctan x-x}{x^2}$ without Taylor's formula or L'Hôpital's rule?

I have to find $$\lim_{x \rightarrow 0^+} \frac{ \arctan(x)-x}{x^2}$$ without Taylor's formula or L'Hôpital's rule. How to tackle it? Any idea is welcome.
1
vote
2answers
207 views

Examples of limits that become easier with Taylor series

There are examples of questions on this site where the OP asks for help solving a limit problem, and some of the answers make use of clever Taylor expansions to evaluate the limit. The purpose of this ...
3
votes
5answers
162 views

Evaluating $\lim_{x\to 0}\frac{\sin(x)\arcsin(x)-x^2}{x^6}$ Step by Step Using L' Hopital Rule

The limit to be found is $$ \lim_{x\to 0}\frac{\sin(x)\arcsin(x)-x^2}{x^6}$$ I've tried l'hopital rule but it gets really messy. I've also tried splitting it into 2 limits but that doesn't work. I ...
4
votes
1answer
186 views

How to calculate $\lim \limits_{x \to 0} \frac{x^2 \sin^2x}{x^2-\sin^2x}$ with $\lim \limits_{x \to 0} \frac{\sin x}{x}=1$?

How to calculate $$\lim \limits_{x \to 0} \frac{x^2 \sin^2x}{x^2-\sin^2x}$$ with $$\lim \limits_{x \to 0} \frac{\sin x}{x}=1?$$ Yes I know the question has been asked, the answer is $3$, L'Hospital ...

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