Discuss the continuity of the function at x 2 where f x is defined by ...........

Mathematics (NDA) limits

Discuss the continuity of the function at x = 2, where f(x) is defined by

f(x) = (2x – 2) / x, for 0 < x ≤ 2

= sin (x – 2) / (x – 2), for x > 2

f is continuous at x =2

f is discontinuous at x = 2

f is differentiable at x=2

none of these

Consider the left- and right-hand limits,

$\lim_{x\rightarrow 2(-)}$ f (x) = $\lim_{x\rightarrow 2}$ (2x- 2) /x

= (2×2 – 2)/ 2 = 1

$\lim_{x\rightarrow 2(+)}$ f(x) = $\lim_{(x-2)\rightarrow 0}$ sin (x-2) / (x-2)

= 1

Therefore, $\lim_{x\rightarrow 2(-)}$ ?f(x) = $\lim_{x\rightarrow 2(+)}$ f(x) = f (2)

Thus, f is continuous at x = 2.

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