How does f(x) = x sin 1/x if x ≠ 0; 0 if x = 0 behave at x = 0?

Question 29

HardOriginal question

How does $f(x) = \begin{cases} x \sin \frac{1}{x}, & x \ne 0 \\ 0, & x = 0 \end{cases}$ behave at $x = 0$ ?

Continuous at the pointCorrect answerYour answerNot 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 answer

Show the answer and explanation

CorrectNot quiteAnswer A

Correct answer: A

$-\lvert x \rvert \le x \sin \frac{1}{x} \le \lvert x \rvert$ , because $\sin \frac{1}{x}$ stays between $-1$ and 1.
Both bounds tend to 0, so $x \sin \frac{1}{x} \to 0 = f(0)$ from either side: continuous.

Worth noting

$\sin \frac{1}{x}$ alone has no limit at 0, but the factor $x$ squeezes the product to 0.

Concept: Testing continuity at a point
Typical timeTypical: 120 s
Your timeYours: —

The rule this testsContinuity at a point

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How does f(x)={xsin⁡1x,x≠00,x=0f(x) = \begin{cases} x \sin \frac{1}{x}, & x \ne 0 \\ 0, & x = 0 \end{cases}f(x)={xsinx1​,0,​x=0x=0​ behave at x=0x = 0x=0?

Explanation

How does $f(x) = \begin{cases} x \sin \frac{1}{x}, & x \ne 0 \\ 0, & x = 0 \end{cases}$ behave at $x = 0$ ?