Calculus I, Chapter 3: The Average Rate of Change and Limits
Previous Section: 3.3 The Limit of a Function
Developing Rules about Limits
The first step in any true mathematical exploration is to identify patterns. Mathematics, after all, is in part the study of patterns. We can begin by studying the limits of problems in increasing difficulty.
Example 1: Evaluate lim┬(x→9)5
Solution:
| x | L |
| 8.9 | 5 |
| 8.99 | 5 |
| 8.999 | 5 |
| 9 | 5 |
| 9.001 | 5 |
| 9.01 | 5 |
| 9.1 | 5 |
Thus, lim┬(x→9)5=5.
It is clear to see that lim┬(x→a)constant=constant no matter the value for a.
Example 2: Evaluate lim┬(x→9)x
Solution:
| x | L |
| 8.9 | 8.9 |
| 8.99 | 8.99 |
| 8.999 | 8.999 |
| 9 | 9 |
| 9.001 | 9.001 |
| 9.01 | 9.01 |
| 9.1 | 9.1 |
So, lim┬(x→9)x=9 .
Example 3: Evaluate lim┬(x→9)〖2x^2 〗
Solution:
| x | L |
| 8.9 | 158.42 |
| 8.99 | 161.6402 |
| 8.999 | 161.964 |
| 9 | 162 |
| 9.001 | 162.036 |
| 9.01 | 162.3602 |
| 9.1 | 165.62 |
For our third example, lim┬(x→9)〖2x^2 〗=162. It seems that for a monomial, the lim┬(x→a)〖f(x)〗=f(a). That is to say, we could have just plugged the number into the function!
Example 4: Evaluate lim┬(x→9)〖2x^2 〗+x+5
Solution:
| x | L |
| 8.9 | 172.32 |
| 8.99 | 176.6302 |
| 8.999 | 175.963 |
| 9 | 176 |
| 9.001 | 176.037 |
| 9.01 | 176.3702 |
| 9.1 | 179.72 |
If we review the previous examples, we can see that this example is each of the first three examples added together and that the answer can be determined by adding the solutions to the previous problems. We can conclude that the limit of a sum is equal to the sum of the limits. That is to say lim┬(x→a)〖f(x)+g(x)〗=lim┬(x→a)〖f(x)〗+lim┬(x→a)〖g(x)〗. It is important to note that is only works if the limits are tending towards the same input, that is to say that the x-values have to be the same! This works for the previous examples because all of them are the limit as x approaches 9.
Example 5: Evaluate lim┬(x→9)5x
Solution:
| x | L |
| 8.9 | 44.5 |
| 8.99 | 44.95 |
| 8.999 | 44.995 |
| 9 | 45 |
| 9.001 | 45.005 |
| 9.01 | 45.05 |
| 9.1 | 45.5 |
Compare this to example 2. This example demonstrates a constant multiple law, which is just a special case of the sum law where each term is the same: lim┬(x→a)〖k∙f(x)〗=k∙lim┬(x→a)〖f(x)〗
The final two laws can be called the product and quotient laws:
lim┬(x→a)〖f(x)g(x)〗=lim┬(x→a)〖f(x)〗∙lim┬(x→a)〖g(x)〗
lim┬(x→a)〖f(x)/(g(x))〗=lim┬(x→a)〖f(x)〗/lim┬(x→a)〖g(x)〗 ,〖as long as lim┬(x→a)〗〖g(x)〗≠0
Example 6: Evaluate lim┬(x→9)〖(2x^2 〗+x+5)(x)
Solution:
This can be rewritten as lim┬(x→9)〖2x^2 〗+x+5 ∙ lim┬(x→9)x. We showed in previous examples that these are equal to 176 and 9, so lim┬(x→9)〖(2x^2 〗+x+5)(x)=176∙9=1584.
Example 7: Evaluate lim┬(x→9)〖(2x^2)/(2x^2+x+5)〗
Solution:
lim┬(x→9)〖(2x^2)/(2x^2+x+5)〗=(lim┬(x→9) 2x^2)/(lim┬(x→9) 2x^2+x+5)
=162/176=81/88
Review
Limits can be evaluated using a chart, and by breaking the limit up into individual parts to solve each term in the limit
Sometimes a limit law will apply which allows us to solve the limit without a chart
Equations
Assuming that each of the limits exist:
Constant Law lim┬(x→a)k=k
Sum Law lim┬(x→a)〖f(x)+g(x)〗=lim┬(x→a)〖f(x)〗+lim┬(x→a)〖g(x)〗.
Constant Multiple Law lim┬(x→a)〖k∙f(x)〗=k∙lim┬(x→a)〖f(x)〗
Product Law lim┬(x→a)〖f(x)g(x)〗=lim┬(x→a)〖f(x)〗∙lim┬(x→a)〖g(x)〗
Quotient Law lim┬(x→a)〖f(x)/(g(x))〗=lim┬(x→a)〖f(x)〗/lim┬(x→a)〖g(x)〗 ,〖as long as lim┬(x→a)〗〖g(x)〗≠0
Substitution lim┬(x→a)〖f(x)〗=f(a)