If x=u is a point of discontinuity of f(x) =limn→∞ cos2n⁡x then the value of cos u is

If $x = u$ is a point of discontinuity of $f (x) =$
$lim_{n \to \infty} \cos^{2 n} x$ then the value of cos $u$ is

A
0
B
$1 / 2$
C
$(- 1)^{n}$
D
1

Solution:

$\begin{array}{l} f (x) = lim_{n \to \infty} {(\cos^{2} x)}^{n} \\ = \{\begin{cases} 0, & 0 \leq \cos^{2} x < 1 \\ 1, & \cos^{2} x = 1 \end{cases} \\ = \{\begin{cases} 0, & x \neq n π \\ 1, & x = n π \end{cases} \end{array}$

Hence $f (x)$ is discontinuous when $u = n π, n \in I$

So, $\cos u = \cos n π = (- 1)^{n}$

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