Question

Evaluate the following limit using continuity:

lim_ x→π ​ cos(xcos(x))
a) cos(π)
b) −cos(π)
c) cos(0)
d) −cos(0)

Answer 1

The limit
`lim x→π cos(xcos(x))` can be evaluated by checking the continuity of the function
`cos(xcos(x)) at x = π`. If the function is continuous the limit will be equal to the function value at
`x = π`, which is
`cos(π)`.

Step-by-step explanation:

To evaluate the limit
`lim x→π cos(xcos(x))` using continuity, we need to check if the function
`cos(xcos(x))` is continuous at
`x = π`.

If it is continuous then the limit will be equal to the function value at

`x = π`.

Let's check if the function
`cos(xcos(x))` is continuous at
`x = π`. Cosine function is continuous everywhere, and the composition of continuous functions is also continuous.

Therefore,
`cos(xcos(x))` is continuous at
`x = π`.

Hence, the limit
`lim x→π cos(xcos(x))` is equal to
`cos(π)`, which is the option (a)
`cos(π).`