1. What Do Limits Have To Do With Calculus? An Unlimited Review of Limits

2. Definition of a Limit The simplest way to understand a limit of a function is by looking at it graphically. Assume we’re looking at this function: Let’s see what its graph looks like and what happens as x approaches 2…

3. 2 y x It appears as the x-coordinates get closer and closer to 2, the y- coordinates get closer and closer to 3. We can say that the limit of this function is 3. A limit is the value a function, f(x), approaches as the variable within that function gets nearer and nearer to a particular value, x. 3

4. Evaluating Limits Analytically Sometimes looking at a graph is not possible, so you need to have other ways to find a limit. We can evaluate limits analytically using algebraic techniques. These are: –Substitution –Factoring (simplifying the expressions) –Rationalizing the numerator or denominator (conjugates)

5. Substitution Example 1: Evaluate the following limit: Notice that you can substitute x = 2 into the function to get 2 2 – 4 *2 + 9 = 5. Therefore, the limit of this function as x approaches 2 is 5. Try this one:

6. Now, if all limits were this easy, you would have worked with them in Algebra I, but…

7. Factoring Sometimes substitution just doesn’t work. Let’s look at Example 2: indeterminate If you tried using substitution, the result would be 0/0. The technical term for a result of 0/0 is indeterminate, which means that you cannot determine the limit using this method. What can we do? Simply do what the title of this slide says: Factor!

8. Using Conjugates The last limit evaluation method attacks the radical expression in limits. This makes applying the conjugate method rather easy. Let’s evaluate Example 3:

9. If you tried substitution, it will result in the indeterminate answer 0/0, and factoring isn’t very helpful. To evaluate this limit, multiply the fraction by the conjugate of the radical expression. But remember you need to multiply by 1, so you must multiply by the conjugate/itself. Let’s look at this:

10. Try These Do not use your calculator for these.

11. Evaluating Limits Numerically By looking at a graph’s table of data, we can determine the limit of the function. Let’s look at Example 4: X2.92.992.9993.0013.013.1 g(x)-4.41-4.9401-4.9945.0065.06015.61 From the above data, estimate the value of the following limits:

12. X2.92.992.9993.0013.013.1 g(x)-4.41-4.9401-4.9945.0065.06015.61

13. Properties of Limits If c, k, R, S, U, and V are finite numbers and if

14. Special Limits

15. Rational Function Theorem Using the fact that… We can determine the limit of a quotient of polynomials (rational functions)

16. Try These Do not use your calculator

18. Answers will be posted later.