Question

Let f be a continuous function such that f(1) = 3/2 and f(1/2^0.5) = 3. Then the value of \[\lim_{x \rightarrow 0}f(\frac{ \sqrt{x+1}-\sqrt{1-x} }{x })\]

Answer 1

@hartnn

Answer 2

Since f is continuous, we must have that
\[ \lim_{x\rightarrow0}f(\frac{\sqrt{x+1}-\sqrt{1-x}}{x})=f(\lim_{x\rightarrow0}\frac{\sqrt{x+1}-\sqrt{1-x}}{x}) \]
Applying L'Hopitals rule to the limit,
\[ \lim_{x\rightarrow0}\frac{\sqrt{x+1}-\sqrt{1-x}}{x}=\lim_{x\rightarrow0}\frac{\frac{1}{2\sqrt{x+1}}-\frac{1}{2\sqrt{1-x}}}{1}=\frac{1}{2\sqrt{0+1}}-\frac{1}{2\sqrt{1-0}}=1 \]
So the initial limit simply is
\[ f(1)=\frac{3}{2} \]

Answer 3

Exactly