Functions
Differentiability
MMTS_Full_Test_17
Grade 12
Question:
$f(x)=\left|x^2-x-6\right|/|x-3|$ at $x=3$. Which is true?
$f$ is continuous and differentiable
$f$ is continuous but not differentiable
$f$ is not continuous
$\lim_{x\to3}f(x)$ does not exist
Step-by-Step Solution
Key Concept: $x^2-x-6=(x-3)(x+2)$; so $f(x)=|x+2|$ for $x\ne 3$
$f(x)=\frac{|x-3||x+2|}{|x-3|}=|x+2|$ for $x\ne3$. $f$ can be continuously extended; but as stated, $f$ is not defined at $x=3$ so not continuous. Answer: 3.
Correct Answer: 3