Linked Questions

1 vote
0 answers
71 views

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 ...
SRobertJames's user avatar
  • 4,268
1 vote
0 answers
66 views

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 ...
Grigor Hakobyan's user avatar
1 vote
0 answers
321 views

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} $...
Ben G's user avatar
  • 893
1 vote
0 answers
40 views

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?
AdamK's user avatar
  • 333
1 vote
0 answers
401 views

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: $$\...
Robert Fitzgerald's user avatar
1 vote
0 answers
794 views

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 $\sin \theta \approx \theta$ $\tan\theta \approx \theta$ $\cos\theta \approx 1-\frac{\theta^2}{2}$ ...
Espantosidad's user avatar
0 votes
0 answers
55 views

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 &...
lingku's user avatar
  • 11
0 votes
0 answers
70 views

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 ...
Pastudent's user avatar
  • 868
0 votes
0 answers
45 views

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\...
Nav Bhatthal's user avatar
  • 1,057
0 votes
0 answers
47 views

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 {\...
santijrv's user avatar
0 votes
0 answers
63 views

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{...
toronto hrb's user avatar
0 votes
0 answers
55 views

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 ...
J.Doe's user avatar
  • 11
0 votes
0 answers
92 views

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 $\forall\epsilon>0 \,\,\exists \delta >0$ ...
user372003's user avatar
0 votes
0 answers
6k views

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 ...
Logic's user avatar
  • 11
0 votes
0 answers
931 views

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 ...
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