Question

Please , continuity , for the function

Answer 1

Answer: D

Explanation:

This is a piecewise function. Normally cos x and tan x are continuous functions on their own. but you are using cos x as part and tan x as other part.

In order to see if the function is continuous at 0 plug in 0 for both functions. If it is the same then it is continuous, if different then it is discontinuous. There are no asymptotes in either of the functions so there wouldn't be in any of your answers

2 cos x

2 cos 0 > use unit circle to solve

2 (1)

2

2 + tan x

2 + 0

2

You get the same answer from both functions and one of the symbols is ≥, so the function is continuous.

D is your answer

Answer 2

The teacher made a mistake. None of the answer choices work because 2+tan(x) has infinitely many vertical asymptotes. Each time there's a vertical asymptote, there's a disconnect.

The answer should be something like
`\text{The function is continuous almost everywhere except}\\\text{when x} = (\pi)/(2)+n\pi, \ \text{where n is an integer and n} \ge 0`

For instance, if n = 0 then x = pi/2 leads to an asymptote.