How does f(x) = x^2/(| x |) if x ≠ 0; 0 if x = 0 behave at x = 0?

Question 24

MediumOriginal question

How does $f(x) = \begin{cases} \frac{x^{2}}{\lvert x \rvert}, & x \ne 0 \\ 0, & x = 0 \end{cases}$ behave at $x = 0$ ?

Not continuous: the left and right limits differCorrect answerYour answerNot continuous: the limit exists but differs from the valueCorrect answerYour answerNot continuous: the function has no value thereCorrect answerYour answerContinuous at the pointCorrect answerYour answer

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CorrectNot quiteAnswer D

Correct answer: D

For $x \ne 0$ , $\frac{x^{2}}{\lvert x \rvert} = \lvert x \rvert$ , since $x^{2} = \lvert x \rvert^{2}$ .
So both one-sided limits are 0, equal to $f(0)$ : continuous.

Worth noting

Unlike $\frac{\lvert x \rvert}{x}$ , this quotient has $x^{2}$ on top, which is positive on both sides: no jump.

Concept: Testing continuity at a point
Typical timeTypical: 90 s
Your timeYours: —

The rule this testsContinuity at a point

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How does f(x)={x2∣x∣,x≠00,x=0f(x) = \begin{cases} \frac{x^{2}}{\lvert x \rvert}, & x \ne 0 \\ 0, & x = 0 \end{cases}f(x)={∣x∣x2​,0,​x=0x=0​ behave at x=0x = 0x=0?

Explanation

How does $f(x) = \begin{cases} \frac{x^{2}}{\lvert x \rvert}, & x \ne 0 \\ 0, & x = 0 \end{cases}$ behave at $x = 0$ ?