Squeeze Theorem for AP Calculus

Practice problems for this concept can be found at Limits and Continuity Practice Problems for AP Calculus.

If f, g, and h are functions defined on some open interval containing a such that g(x) ≤ f(x) ≤ h(x) for all x in the interval except possibly at a itself, and

Theorems on Limits

Example 1

Find the limit if it exists:

Substituting 0 into the expression would lead to 0/0. Rewrite and thus, . As x approaches 0, so does 3x. Therefore, (Note that is equivalent to by replacing 3x by x.) Verify your result with a calculator. (See Figure 5.1-7.)

Example 2

Find the limit if it exists:

Rewrite As h approaches 0, so do 3h and 2h. Therefore, (Note that substituting h = 0 into the original expression would have produced 0/0). Verify your result with a calculator. (See Figure 5.1-8.)

Example 3

Find the limit if it exists:

Substituting 0 in the expression would lead to 0/0. Multiplying both the numerator and denominator by the conjugate (1 + cos y) produces

Example 4

Find the limit if it exists:

Using the quotient rule for limits, you have Verify your result with a calculator. (See Figure 5.1-10.)

Practice problems for this concept can be found at Limits and Continuity Practice Problems for AP Calculus.