Question

True/False: if f'(a) exists then the limit as x approaches a f(x) is equal to f(a)

Answer 1

False

Answer 2

If \(f'(a)\) exists, then \(\displaystyle \lim_{x\to a}f(x)=f(a)\).

Counterexample: \(\displaystyle f(x)=\left|x-a\right| \)

Note that if you plot the function, there won't be a tangent at \(x=a\).
(Can't be tangent to two perpendicular lines simultaneously.)

Also, if you differentiate you get:
\(\displaystyle f'(x)=\frac{d}{dx}\left(\left|x-a\right|\right)=\frac{x-a}{|x-a|}\)

\(\displaystyle f'(a)=\frac{a-a}{|a-a|}=0/0=~\color{red}{\large \bf ???}\)

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Additional material: Absolute value function; derivative;
(Formula derivation)

\(\displaystyle y=|f(x)| \)

\(\displaystyle y=\sqrt{[f(x)]^2} \) (definition of absolute value (for real numbers) \(\displaystyle |z|=\sqrt{z^2} \))

\(\displaystyle y'=\frac{1}{2\sqrt{[f(x)]^2}}\times 2[f(x)]^{2-1}\times f'(x)\) (power rule, and chain rule)

\(\displaystyle y=\frac{f(x)}{|f(x)|}\times f'(x) \) (simplifying, and applying the def. of absolute value)

you will see that this formula is exactly what I applied.

Answer 3

(Note that the hypothesis does meet, and the conclusion does not.) So, you know there is at least one example for which your statement is not true.