Question

If \[\lim_{h \rightarrow 0} \frac{ \arcsin(a+h)-\arcsin(a) }{ h } = 2\] , which of the following could be the value of a?
A) sqrt2/2 B) sqrt3/2 C)sqrt3 D)1/2 E) 2

Answer 1

The left side is straight from the definition of the derivative:
\[\lim_{h\to0}\frac{f(x+h)-f(x)}{h}=:f'(x)\]
As for values of the derivative, you have
\[\lim_{x\to c}\frac{f(x)-f(c)}{x-c}=:f'(c)\]
so you have \(f(x)=\arcsin x\) and (\(c=a\) in this problem).

Recall the derivative of the inverse sine function:
\[f(x)=\arcsin x~~\implies~~f'(x)=\frac{1}{\sqrt{1-x^2}}\]
and so given that \(f'(a)=2\), you have
\[\frac{1}{\sqrt{1-a^2}}=2\]