Solving the Epsilon-Delta Problem with Sinusoidal Functions

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What is the limit of sin(5x)/x as x goes to 0? Taking "A" as that value you want to find x such that |sin(x)/x- A|< 0.01.
 
The limit as sin(5x)/x goes to 0 is 5, right?
 
as a practical matter you may want to find t such that:

|sin(t)/t - 1| < 0.01

then use x = t/5.

finding such a t (and thus x) isn't that hard (i did it by trial-and-error in about 2 minutes), the real trick is showing the inequality holds in the interval (-x,x). unless i am mistaken, the problem doesn't ask you to find the largest possible such x, just one that works.
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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