Answer
(a) The graph is shown below.
(b) $\lim_{x\to1^-}f(x)=\lim_{x\to1^+}f(x)=1$
(c) $\lim_{x\to1}f(x)=1$
Work Step by Step
(a) The graph is shown below.
(b) Looking at the graph, as $x$ approaches $1$ from both the left and the right, $f(x)$ would arbitrarily get close to $1$. So, $\lim_{x\to1^-}f(x)=\lim_{x\to1^+}f(x)=1$
- Check algebraically:
$\lim_{x\to1^-}f(x)=\lim_{x\to1^-}(x^3)=1^3=1$
$\lim_{x\to1^+}f(x)=\lim_{x\to1^+}(x^3)=1^3=1$
(c) Since $\lim_{x\to1^-}f(x)=\lim_{x\to1^+}f(x)=1$, as $x$ approaches $1$ from either side, $f(x)$ would still approach $1$.
Therefore, $\lim_{x\to1}f(x)=1$
