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: Using the first-principles definition:
\(f\) is continuous at \(x=c\) if \(\displaystyle\lim_{x\to c}f(x)=f(c)\).

Step 2: Applying limit algebra:
Since \(\sin x\) and \(\cos x\) are continuous, \(\displaystyle\lim_{x\to c}\sin x=\sin c\) and \(\displaystyle\lim_{x\to c}\cos x=\cos c\).

Step 3: Limit of the quotient:
By the quotient law for limits (valid since \(\cos c\ne0\) in the domain), \(\displaystyle\lim_{x\to c}\dfrac{\sin x}{\cos x}=\dfrac{\sin c}{\cos c}=\tan c=f(c)\).

Final Answer:
\[ \boxed{\tan x \text{ is continuous at every } c \text{ in its domain}} \]