Question

Prove that \(f(x)=\tan x\) is a continuous function.

Hint:

tan x = sin x / cos x is a quotient of continuous functions, continuous wherever cos x ≠ 0.

Answer 1

Step 1: Writing tan x as a quotient:
\(f(x)=\tan x=\dfrac{\sin x}{\cos x}\).

Step 2: Continuity of sin x and cos x:
Both \(\sin x\) and \(\cos x\) are known to be continuous for every real \(x\) (standard results).

Step 3: Quotient rule for continuity:
The quotient of two continuous functions is continuous at every point where the denominator is non-zero.

Final Answer:
Since \(\cos x\ne0\) throughout the domain of \(\tan x\), \(f(x)=\tan x\) is continuous on its domain.\[ \boxed{\tan x \text{ is continuous wherever } \cos x\ne 0} \]