Linked Questions
289 questions linked to/from How to prove that $\lim\limits_{x\to0}\frac{\sin x}x=1$?
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Geometric proof that $\tan \alpha \geq \alpha$ for any acute angle $\alpha$
For any acute angle $\alpha$, then $\tan \alpha \geq \alpha$. Is it possible to prove this geometrically?
My work so far:
A rigorous proof requires defining a radian measure of angles: this can be ...
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Why does $\sin(\frac{1}{x})$ give reciprocals? [duplicate]
I was preparing a calculus lesson for my students where we examine the continuity of various functions. One such function was
$$g(x) = \begin{cases}
0 & x = 0\\
x \sin(\frac{1}{x})) & x \neq 0 ...
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Geometric Intuition for limit of 1-cos(x)/x as x->0
I've worked my way through the geometric Squeeze Theorem proof of $\lim\limits_{x\to 0}\frac{\sin{x}}{x} $.
I was wondering how the same could be done for:
$$\lim\limits_{x\to 0}\frac{1 - \cos{x}}{x} $...
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Prove trig inequality [duplicate]
I know that $\cos x \leq (\sin x)/x \leq 1/(\cos x)$, but I need to prove this and I don't know how to do that. I've seen diagrams used to show the relationship, but can it be done without?
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Area of a Circle using Squeeze Theorem
Someone showed me a derivation for the area of a circle today. They took a circle of radius $r$ and inscribed a regular polygon in the circle. If you take an $n$-sided polygon, then its area is:
$$\...
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A geometric proof for the "small angle approximation" for the sine, cosine and tangent
How can be deduced the so called "small angle approximations" for sine, cosine and tangent, namely
$\sin \theta \approx \theta$
$\tan\theta \approx \theta$
$\cos\theta \approx 1-\frac{\theta^2}{2}$
...
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Epsilon delta proof with log and sin
I've been attempting to prove this limit (again). It must be a proof with epsilon delta definition.
$\frac{\sin(\log(x))}{\log(x)} \rightarrow 1$ when $x \rightarrow 1$
so far I have, given $\epsilon &...
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proof that $\sin(x-y)/(x-y)$ is continuous.
The question was implicitely already discussed in this Post.
However, the only answer which used an argument similar to my proofs isn't very detailed and I was wondering whether or not my proofs ...
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Derivative of sinx from first principle [duplicate]
I am just confused over this step in the derivative of sin x
$$
\lim_{x \to 0}{\sin\left(x\right) \over x} = 1
$$
When we have the limit, my textbook uses the small angle approximations, to say $\sin\...
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How can prove limit of $ n\sin(\frac{1}{n})=1 $ as $ n $ approaches to infinity? [duplicate]
I am trying to solve this sequence limit
$\lim_{n\to \infty} $ $n\sin(\frac{1}{n})=1$
and I’m struggling finding a bound for $|a_n -l|$
I thought of rewriting
$|n\sin(\frac{1}{n}) - 1|$
As
$|\frac {\...
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Prove $\lim_{x\rightarrow c} \sin x =\sin c$ using $\epsilon$-$\delta$ definition [duplicate]
Proof. $\forall \epsilon>0$, choose $\delta=\epsilon$. Now $\forall x$, if $0<|x-c|<\delta$, then $|\sin x -\sin c|=|2\cos\frac{x+c}{2} \sin\frac{x-c}{2}|\le 2|\sin \frac{x-c}{2}|\le 2|\frac{...
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I want to know rigorous proof of lim x→0 sin(x)/x=1 [duplicate]
we gets (sinx)'=cosx from lim x→0 sin(x)/x=1
Geometric proof of it is based on the fact that the area of sector of unit circle with central angle x is x/2
However, when we calculate the area of unit ...
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How would I prove this limit?
Prove using the definition of a limit that $$\lim_\limits{x\to0}\sin(x)=0. $$
And $$\lim_\limits{x\to0}\frac{\sin(x)}{x}=1.$$
I need to prove that $\forall\epsilon>0 \,\,\exists \delta >0$ ...
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How do I prove that the limit as x approaches 0 of (sin(x)/x) = 1 using the definition of a limit? [duplicate]
If I let r be a positive number, I have to show that there exists a positive number, u, such that |x| < u implies that |(sin(x))/x - 1| < r. Now I've gone about this two different ways, but hit ...
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Prove that these two definitions are equivalent
While answering this question I have used that \begin{equation}\sin x=\displaystyle\sum_{n=0}^\infty\dfrac{(-1)^nx^{2n+1}}{(2n+1)!}\end{equation}
Nwe my question is that how can it be shown that the ...