Section 1.3 – Evaluating Limits Analytically

1. Section 1.3 – Evaluating Limits Analytically

2. Direct Substitution One of the easiest and most useful ways to evaluate a limit analytically is direct substitution (substitution and evaluation): If you can plug c into f(x)and generate a real number answer in the range of f(x), that generally implies that the limit exists (assuming f(x) is continuous at c). Example: Always check for substitution first. The slides that follow investigate why Direct Substitution is valid.

3. Properties of Limits Let b and c be real numbers, let n be a positive integer, and let f and g be functions with the following limits:

4. Properties of Limits Let b and c be real numbers, let n be a positive integer, and let f and g be functions with the following limits:

5. Example Let and . Find the following limits.

6. Example 2 Sum/Difference Property Multiple and Constant Properties Direct Substitution Power Property Limit of x Property

7. Direct Substitution • Direct substitution is a valid analytical method to evaluate the following limits. • If p is a polynomialfunction and c is a real number, then: • If r is a rational function given by r(x) = p(x)/q(x), and c is a real number, then • If a radical function where n is a positive integer. The following limit is valid for all c if n is odd and only c>0 when n is even:

8. Direct Substitution • Direct substitution is a valid analytical method to evaluate the following limits. • If the f and g are functions such that Then the limit of the composition is: • If c is a real number in the domain of atrigonometric function then:

9. Example 1 Find if , and if f and g are continuous functions.

10. Example 2 For what value(s) of x can the limit not be evaluated using direct substitution? Direct Substitution can be used since the function is well defined at x=3 At x=-6 since it makes the denominator 0:

11. Indeterminate Form Evaluate the limit analytically: An example of an indeterminate form because the limit can currently not be determined. 1/0 is NOT indeterminate. Often limits can not be evaluated at a value using Direct Substitution. If this is the case, try to find another function that agrees with the original function except at the point in question. In other words… Note: If direct substitution results in 0/0 (or other indeterminates: ∞/∞,∞x0, ∞-∞), the limit probably exists. How can we simplify: ?

12. Strategies for Finding Limits • To find limits analytically, try the following: • Direct Substitution (Try this FIRST) • If Direct Substitution fails, then rewrite then find a function that is equivalent to the original function except at one point. Then use Direct Substitution. Methods for this include… • Factoring/Dividing Out Technique • Rationalize Numerator/Denominator • Eliminating Embedded Denominators • Trigonometric Identities • Legal Creativity

13. Example 1 Evaluate the limit analytically: Factor the numerator and denominator At first Direct Substitution fails because x=2 results in 0/0. (Remember that this means the limit probably exists.) Cancel common factors This function is equivalent to the original function except at x=2 Direct substitution

14. Example 2 Evaluate the limit analytically: Rationalize the numerator Cancel common factors Direct substitution

15. Example 3 Evaluate the limit analytically: Cancel the denominators of the fractions in the numerator If the subtraction is backwards, Factoring a negative 1 to flip the signs Cancel common factors Direct substitution

16. Example 4 Evaluate the limit analytically: Expand the the expression to see if anything cancels Factor to see if anything cancels Direct substitution

17. Example 5 Evaluate the limit analytically: Eliminate the embedded fraction Rewrite the tangent function using cosine and sine If the subtraction is backwards, Factoring a negative 1 to flip the signs Direct substitution

18. Two “Freebie” Limits The following limits can be assumed to be true (they will be proven later in the year) to assist in finding other limits: Use the identities to help with these limits. They are located on the first page of your textbook.

19. Example 1 Evaluate the limit analytically: If 3x is the input of the sine function then 3x needs to be in the denominator Isolate the “freebie” Scalar Multiple Property Assumed Trig Limit

20. Example 2 Evaluate the limit analytically: Try multiplying by the reciprocal Use the Trigonometry Laws Split up the limits A freebie limit and Direct substitution