Find Limits of Composition in the graph of $f$

Question

The graph of the function $f$ is shown in the figure above. The value of $ \lim_\limits{x\to 0} f(1-x^2) $ is

When I try to solve this, I get $f(1)$, which should equal $2$. However, the answer key states that it should equal $3$.

Answer 1

Note that for $x\approx 0$ we have that

• $1-x^2\le 1$

and $1-x^2\to 1$ as $x\to 0$ then

$$\lim_{x\to 0} f(1-x^2)=\lim_{x\to 1^-} f(x)$$

Note also that the function is not continuos at $x=1$ snce

$$\lim_{x\to 1^-} f(x)\neq \lim_{x\to 1^+} f(x)\neq f(1)$$