31. Show that the function defined byf (x) = \cos (x^2 ) is a continuous function.

Answers (1)

Given function is
f (x) = \cos (x^2 )
given function is defined for all real values of x
Let x = k + h
if x\rightarrow k , \ then \ h \rightarrow 0
f(k) = \cos k^2\\ \lim_{x \rightarrow k}f(x) = \lim_{x \rightarrow k}\cos x^2 = \lim_{h \rightarrow 0}\cos (k+h)^2 = \cos k^2\\ \lim_{x \rightarrow k}f(x) = f(k)
Hence, the function  f (x) = \cos (x^2 ) is a continuous function

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