f(x) = x2 sin(1/x), if x is not equal to 0 0, if x = 0 Discuss the continuity of - Maths - Continuity and Differentiability

Question

f(x) = x2 sin(1/x), if x is not equal to 0
0, if x = 0
Discuss the continuity of the function at x = 0

Lalit Mehra · Accepted Answer

The given function is $f (x) = {\begin{matrix} x^{2} \sin (\frac{1}{x}), & x \neq 0 \\ 0, & x = 0 \end{matrix}$

Consider the LHL of f(x) at x = 0.

Put x = 0 – h

When x → 0, h → 0

$∴ \lim_{x \to 0} f (x) = \lim_{h \to 0} (0 - h)^{2} \sin (\frac{1}{0 - h}) = - \lim_{h \to 0} h^{2} \sin (\frac{1}{h}) (∵ \sin (- θ) = - \sin θ) = - 0 \times (A finite value lying between - 1 and 1) = 0$

Consider the RHL of f(x) at x = 0.

Put x = 0 + h

When x → 0, h → 0

$∴ {\lim_{x \to 0}}_{+} f (x) = \lim_{h \to 0} (0 + h)^{2} h (\frac{1}{0 + h}) = \lim_{h \to 0} h^{2} \sin (\frac{1}{h}) = 0 \times (A finite value between – 1 and 1) = 0$

Given, f (0) = 0

$Since, \lim_{x \to 0^{-}} f (x) = \lim_{x \to 0^{+}} f (x) = f (0)$

Hence, f(x) is continuous at x = 0.

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f(x) = x2 sin(1/x), if x is not equal to 0 0, if x = 0 Discuss the continuity of the function at x = 0.

f(x) = x² sin(1/x), if x is not equal to 0

0, if x = 0

Discuss the continuity of the function at x = 0.