Assertion (A): No value of k makes f(x) = (| x |)/x for x ≠ 0, f(0) = k, continuous at 0. Reason (R): The left and right limits of (| x |)/x at 0 are -1 and 1.

Question 36

MediumOriginal question

Assertion (A): No value of $k$ makes $f(x) = \frac{\lvert x \rvert}{x}$ for $x \ne 0$ , $f(0) = k$ , continuous at 0.
Reason (R): The left and right limits of $\frac{\lvert x \rvert}{x}$ at 0 are $-1$ and 1.

Both A and R are true, and R is the correct explanation of A.Correct answerYour answerBoth A and R are true, but R is not the correct explanation of A.Correct answerYour answerA is true, but R is false.Correct answerYour answerA is false, but R is true.Correct answerYour answer

Show the answer and explanation

CorrectNot quiteAnswer A

Correct answer: A

A is true: no single value can match two different one-sided limits.
R is true and is the reason: with no limit at 0, choosing $f(0)$ cannot help.

Concept: When no constant works
Typical timeTypical: 90 s
Your timeYours: —

The rule this testsChoosing a constant for continuity

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Ten questions from Continuity & Differentiability, at your own pace.

Assertion (A): No value of kkk makes f(x)=∣x∣xf(x) = \frac{\lvert x \rvert}{x}f(x)=x∣x∣​ for x≠0x \ne 0x=0, f(0)=kf(0) = kf(0)=k, continuous at 0.Reason (R): The left and right limits of ∣x∣x\frac{\lvert x \rvert}{x}x∣x∣​ at 0 are −1-1−1 and 1.

Explanation

Assertion (A): No value of $k$ makes $f(x) = \frac{\lvert x \rvert}{x}$ for $x \ne 0$ , $f(0) = k$ , continuous at 0.
Reason (R): The left and right limits of $\frac{\lvert x \rvert}{x}$ at 0 are $-1$ and 1.