How does f(x) = (sin 3x)/x if x ≠ 0; 1 if x = 0 behave at x = 0?

Question 26

HardOriginal question

How does $f(x) = \begin{cases} \frac{\sin 3x}{x}, & x \ne 0 \\ 1, & x = 0 \end{cases}$ behave at $x = 0$ ?

Not continuous: the limit exists but differs from the valueCorrect answerYour answerContinuous at the pointCorrect answerYour answerNot continuous: the left and right limits differCorrect 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 $3$ , the right limit $3$ and $f(0) = 1$ .

Worth noting

$\frac{\sin 3x}{x} \to 3$ , not 1: the 3 inside the sine comes out.

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)={sin⁡3xx,x≠01,x=0f(x) = \begin{cases} \frac{\sin 3x}{x}, & x \ne 0 \\ 1, & x = 0 \end{cases}f(x)={xsin3x​,1,​x=0x=0​ behave at x=0x = 0x=0?

Explanation

How does $f(x) = \begin{cases} \frac{\sin 3x}{x}, & x \ne 0 \\ 1, & x = 0 \end{cases}$ behave at $x = 0$ ?