Get Beginner Calculus Help: Solving Limit Problems with Ease

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I realize this question will probably be too easy for people who are good at it.

But I can't figure this out.

I'm trying to find the limit for

lim x→7 (sin(x − 7))/(x2 + 2x − 63)

Sorry, I don't know how to make it look like the proper equation.

Thanks for any help...
 
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welcome to PF!

now what have you tried, or what relevant equations/theorems do you know?
 

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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