How to evaluate this limit?

Question

((((((((https://drive.google.com/file/d/1I7an_KPDQdQ1hcOiBluEnMFNLoQk35dw/view?usp=drivesdk)))))))

How is that equal to 1.

Answer 1

This looks scary. So let's just write what is in the parenthesis and denominator as g(x)

Then this becomes

LIm sin g(x) / g(x) . Since sin(0) = 0 we can apply lhopitals, giving

= lim [ cos(g(x)) g'( x) ] / g'(x) ( by chain rule. Note the derivatives cancel)

= cos (0) = 1

Hope that helps

Patrick