Limits Involving Trigonometric Functions

The trigonometric functions sine and cosine have four important limit properties:


 

You can use these properties to evaluate many limit problems involving the six basic trigonometric functions.

Example 1: Evaluate .

Substituting 0 for x, you find that cos x approaches 1 and sin x − 3 approaches −3; hence,


 


Example 2: Evaluate

Because cot x = cos x/sin x, you find The numerator approaches 1 and the denominator approaches 0 through positive values because we are approaching 0 in the first quadrant; hence, the function increases without bound and and the function has a vertical asymptote at x = 0.

Example 3: Evaluate

Multiplying the numerator and the denominator by 4 produces


 


Example 4: Evaluate .

Because sec x = 1/cos x, you find that