Calculating Limits Using the Limit Laws

Calculating Limits Using the Limit Laws

• In this section we use the following properties of limits, called the Limit Laws, to calculate limits.

Calculating Limits Using the Limit Laws

• These five laws can be stated verbally as follows:

• Sum Law 1. The limit of a sum is the sum of the limits.

• Difference Law 2. The limit of a difference is the difference of the limits.

• Constant Multiple Law 3. The limit of a constant times a function is the constant times the limit of the function.

Calculating Limits Using the Limit Laws

• Product Law 4. The limit of a product is the product of the limits.

• Quotient Law 5. The limit of a quotient is the quotient of the limits (provided that the limit of the denominator is not 0).

• For instance, if f (x) is close to L and g (x) is close to M, it is reasonable to conclude that f (x) + g (x) is close to L + M.

Example 1

• Use the Limit Laws and the graphs of f and g in Figure 1 to evaluate the following limits, if they exist.

Example 1(a) – Solution

• From the graphs of f and g we see that

• and

• Therefore we have

Example 1(b) – Solution

• We see that limx 1 f (x) = 2. But limx  1 g (x) does not exist because the left and right limits are different:

• So we can’t use Law 4 for the desired limit. But we can use Law 4 for the one-sided limits:

• The left and right limits aren’t equal, so limx  1 [f (x)g (x)] does not exist.

Example 1(c) – Solution

• The graphs show that

• and

• Because the limit of the denominator is 0, we can’t use Law 5.

• The given limit does not exist because the denominator approaches 0 while the numerator approaches a nonzero number.

Calculating Limits Using the Limit Laws

• If we use the Product Law repeatedly with g(x) = f (x), we obtain the following law.

• In applying these six limit laws, we need to use two special limits:

• These limits are obvious from an intuitive point of view (state them in words or draw graphs of y = c and y = x).

Calculating Limits Using the Limit Laws

• If we now put f (x) = x in Law 6 and use Law 8, we get another useful special limit.

• A similar limit holds for roots as follows.

• More generally, we have the following law.

Calculating Limits Using the Limit Laws

• Functions with the Direct Substitution Property are called continuous at a.

• In general, we have the following useful fact.

Calculating Limits Using the Limit Laws

• Some limits are best calculated by first finding the left- and right-hand limits. The following theorem says that a two-sided limit exists if and only if both of the one-sided limits exist and are equal.

• When computing one-sided limits, we use the fact that the Limit Laws also hold for one-sided limits.

Calculating Limits Using the Limit Laws

Calculating Limits Using the Limit Laws

• The Squeeze Theorem, which is sometimes called the Sandwich Theorem or the Pinching Theorem, is illustrated by Figure 7.

• It says that if g (x) is squeezed between f (x) and h (x) near a, and if f and h have the same limit L at a, then g is forced to have the same limit L at a.