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