Key to "Analyze the continuity of the function below. Determine..."

Haven

Answered question

2021-11-05

Analyze the continuity of the function below. Determine the type(s) of discontinuity and justify your answer.
$f (x) = {\begin{cases} x, & i f & x < 0 \\ 3, & i f & x = 0 \\ x^{2}, & i f & 0 < x < 3 \\ x + 6, & i f & x \geq 3 \end{cases}$

Answer & Explanation

wheezym

Skilled2021-11-06Added 103 answers

To determine types of dicontinuity and justify your answer:
$f (x) = {\begin{cases} x, & i f & x < 0 \\ 3, & i f & x = 0 \\ x^{2}, & i f & 0 < x < 3 \\ x + 6, & i f & x \geq 3 \end{cases}$
When x=0
$lim_{x \to 0^{-}} f (x) = lim_{x \to 0^{-}} x = 0$
$lim_{x \to 0^{+}} f (x) = lim_{x \to 0^{+}} x^{2} = 0$
$f (0) = 3$
$lim_{x \to 0} f (x) \neq f (0)$
f(x) is not continuous at x=0
when x=3
$lim_{x \to 3^{-}} f (x) = lim_{x \to 3^{-}} x^{2} = 9$
$lim_{x \to 3^{+}} f (x) = lim_{x \to 3^{+}} x + 6 = 9$
$f (3) = 3 + 6 = 9$
$lim_{x \to 3} f (x) = f (3)$
f(x) is continuous at x=3
Therefore, f(x) is not continuous only at x=0

Jeffrey Jordon

Expert2022-06-28Added 2605 answers

Answer is given below (on video)

Do you have a similar question?

Recalculate according to your conditions!

New Questions in Calculus 1

Didn't find what you were looking for?