Determine whether f'(0) exists. f(x)=x2sin⁡1x if x is not equal to 0, 0 if x=0

Recall that:f′(a)=limx⇒af(x)−f(a)x−af′(0)=limx⇒0x2sin⁡(1x)−0x−0f′(0)=limx⇒0xsin⁡(1x)Squeeze theoremIfh(x)≤f(x)≤g(x)Andlimx⇒ah(x)=LAndlimx⇒ag(x)=LThenlimx⇒af(x)=LSince −1≤sin⁡(1x)≤1Therefore −x≤xsin⁡(1x)≤xSincelimx⇒0−x=0Andlimx⇒0x=0By Squeeze Theorem, We havelimx⇒0xsin⁡(1x)=0Result:f'(0)=0

Answered: "Determine whether f'(0) exists. f(x)=x2sin⁡1x if x is..."

Leonidas Cook

Open question

2022-08-18

Determine whether f'(0) exists.

f (x) = x^{2} \frac{\sin 1}{x}

if x is not equal to 0, 0 if x=0

Answer & Explanation

betoosolis7i

Beginner2022-08-19Added 12 answers

Recall that:

f^{'} (a) = lim_{x \Rightarrow a} \frac{f (x) - f (a)}{x - a}

f^{'} (0) = lim_{x \Rightarrow 0} \frac{x^{2} \sin (\frac{1}{x}) - 0}{x - 0}

f^{'} (0) = lim_{x \Rightarrow 0} x \sin (\frac{1}{x})

Squeeze theorem
If

h (x) \leq f (x) \leq g (x)

And

lim_{x \Rightarrow a} h (x) = L

And

lim_{x \Rightarrow a} g (x) = L

Then

lim_{x \Rightarrow a} f (x) = L

Since

- 1 \leq \sin (\frac{1}{x}) \leq 1

Therefore

- x \leq x \sin (\frac{1}{x}) \leq x

Since

lim_{x \Rightarrow 0} - x = 0

And

lim_{x \Rightarrow 0} x = 0

By Squeeze Theorem, We have

lim_{x \Rightarrow 0} x \sin (\frac{1}{x}) = 0

Result:
f'(0)=0

Do you have a similar question?

Recalculate according to your conditions!

New Questions in Differential Calculus

What is the Mixed Derivative Theorem for mixed second-order partial derivatives? How can it help in calculating partial derivatives of second and higher orders?
How do I find the y-intercept of a parabola?
What are the vertices of $9 x^{2} + 16 y^{2} = 144$ ?
How to determine the rate of change of a function?
Why are the tangents for 90 and 270 degrees undefined?
How to find the center and radius of the circle $x^{2} + y^{2} - 6 x + 8 y = 0$ ?
What is multiplicative inverse of a number?
How to find the continuity of a function on a closed interval?
How do I find the tangent line of a function?
How to find vertical asymptotes using limits?
How to find the center and radius of the circle $x^{2} - 12 x + y^{2} + 4 y + 15 = 0$ ?
Let f be a function so that (below). Which must be true?
I. f is continuous at x=2
II. f is differentiable at x=2
III. The derivative of f is continuous at x=2
(A) I (B) II (C) I and II (D) I and III (E) II and III
How to find the center and radius of the circle given $x^{2} + y^{2} + 8 x - 6 y = 0$ ?
How to find the center and radius of the circle $x^{2} + y^{2} + 4 x - 8 y + 4 = 0$ ?
How to identify the center and radius of the circle ${(x + 3)}^{2} + {(y - 8)}^{2} = 16$ ?

Didn't find what you were looking for?