## 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