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

Question 15

MediumOriginal question

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

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

Show the answer and explanation

CorrectNot quiteAnswer A

Correct answer: A

Continuity at $a$ needs the left limit, the right limit and $f(a)$ to be equal.
Here the left limit is $-1$ , the right limit $1$ and $f(0) = 0$ .

Worth noting

Approaching from the left gives $-1$ , from the right 1: there is no single limit.

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

Explanation

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