1. 2.3 Calculating Limits Using the Limit Laws
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2. Calculating Limits Using the Limit Laws
In this section we use the following properties of limits, called the Limit Laws, to calculate limits.

3. 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.

4. 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.

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

6. Example 1(a) – Solution From the graphs of f and g we see that and
Therefore we have
(by Law 1)
(by Law 3)

7. Example 1(b) – Solution
cont’d
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.

8. Example 1(c) – Solution The graphs show that and
cont’d
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.
Figure 1

9. 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).
Power Law

10. 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.
Root Law

11. 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.
Figure 7

12. Example 1 – Evaluating Basic Limits

13. Example 2 – The Limit of a Polynomial
Solution:

14. Properties of Limits The limit (as x → 2 ) of the polynomial function
p(x) = 4x2 + 3 is simply the value of p at x = 2.
This direct substitution property is valid for all polynomial
and rational functions with nonzero denominators.

15. Example 3 – The Limit of a Rational Function
Find the limit:
Solution:
Because the denominator is not 0 when x = 1, you can
apply Theorem 1.3 to obtain

16. Example 4 – Finding the Limit of a Function
Find the limit:
Solution:
Let f (x) = (x3 – 1) /(x – 1)
By factoring and dividing out like factors, you can rewrite f as

17. f and g agree at all but one point
Example 4 – Solution
cont’d
So, for all x-values other than x = 1, the functions f and g agree, as shown in Figure 1.17
f and g agree at all but one point
Figure 1.17

18. Example 4 – Solution Because exists, you can apply Theorem 1.7 to
cont’d
Because exists, you can apply Theorem 1.7 to
conclude that f and g have the same limit at x = 1.

19. Example 5 – Dividing Out Technique
Find the limit:
Solution:
Although you are taking the limit of a rational function, you cannot apply Theorem 1.3 because the limit of the denominator is 0.

20. Example 5 – Solution
cont’d
Because the limit of the numerator is also 0, the numerator and denominator have a common factor of (x + 3).
So, for all x ≠ –3, you can divide out this factor to obtain
Using Theorem 1.7, it follows that

21. Example 5 – Solution This result is shown graphically in Figure 1.18.
cont’d
This result is shown graphically in Figure 1.18.
Note that the graph of the function f coincides with the graph of the function g(x) = x – 2, except that the graph of f has a gap at the point (–3, –5).
Figure 1.18

22. Rationalizing Technique
Another way to find a limit analytically is the rationalizing technique, which involves rationalizing the numerator of a fractional expression.
Recall that rationalizing the numerator means multiplying the numerator and denominator by the conjugate of the numerator.
For instance, to rationalize the numerator of
multiply the numerator and denominator by the conjugate of
which is

23. Example 6 – Rationalizing Technique
Find the limit:
Solution:
By direct substitution, you obtain the indeterminate form 0/0.

24. Example 6 – Solution
cont’d
In this case, you can rewrite the fraction by rationalizing the numerator.

25. Example 6 – Solution
cont’d
Now, using Theorem 1.7, you can evaluate the limit
as shown.