Section 1.3 – Evaluating Limits Analytically

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.

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: Constant Function Limit of x Limit of a Power of x Scalar Multiple

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: Sum Difference Product Quotient Power

Example Let and. Find the following limits.

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

Direct substitution is a valid analytical method to evaluate the following limits. If p is a polynomial function 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: 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 a trigonometric function then: Direct Substitution

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

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

Indeterminate Form 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… How can we simplify: ? Evaluate the limit analytically: Note: If direct substitution results in 0/0 (or other indeterminates: ∞/∞, ∞ x 0, ∞-∞), the limit probably exists.

Strategies for Finding Limits To find limits analytically, try the following: 1.Direct Substitution (Try this FIRST) 2.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

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

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

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 Direct substitution Cancel common factors

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

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

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.

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

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