Assertion: f(x)=xn sin ((1/x)) is differentiable for all real values of x(n ≥ 2). Reason: For n ≥ 2, displaystyle lim x arrow 0 f(x)=0

Question

Assertion: f(x)=xn sin ((1/x)) is differentiable for all real values of x(n ≥ 2). Reason: For n ≥ 2, displaystyle lim x arrow 0 f(x)=0 (A) Assertion is correct, reason is correct; reason is a correct explanation for assertion. (B) Assertion is correct, reason is correct; reason is not a correct explanation for assertion (C) Assertion is correct, reason is incorrect (D) Assertion is incorrect, reason is correct.

Tardigrade Admin · Accepted Answer

displaystyle lim x arrow 0 f(x)= displaystyle lim x arrow 0 xn sin ((1/x))=0 for positive integer n Now f(0) does not exist, hence function is not continuous at x=0

Q. Assertion : $f (x) = x^{n} sin (\frac{1}{x})$ is differentiable for all real values of $x (n \geq 2)$ .
Reason : For $n \geq 2, x \to 0 lim f (x) = 0$

Assertion is correct, reason is correct; reason is a correct explanation for assertion.

Assertion is correct, reason is correct; reason is not a correct explanation for assertion

Assertion is correct, reason is incorrect

Assertion is incorrect, reason is correct.

$x \to 0 lim f (x) = x \to 0 lim x^{n} sin (\frac{1}{x}) = 0$ for positive integer $n$
Now $f (0)$ does not exist, hence function is not continuous at $x = 0$

Q. Assertion : f(x)=xnsin(x1​) is differentiable for all real values of x(n≥2). Reason : For n≥2,x→0lim​f(x)=0