Evaluating Limits Analytically

1. Evaluating Limits Analytically • Direct Substitution • Cancellation Technique • Rationalization Technique • Squeeze Theorem – not covered

2. If f(x) is defined and c, and, then the limit can be evaluated by direct substitution. The limits of the following functions can be evaluated using direct substitution, so long as domain of f : Direct Substitution • Polynomial functions, including constant, linear and higher degree functions • Rational functions, • Radical Functions (Note, if n is even, ) • Trig functions (recall that quotient & reciprocal trig fcns have domain restrictions) Polynomials are defined over all x.

3. Direction Substitution Examples All of these functions are defined at the provided values of x. So direct substitution can be used: = 22 + 3 = 7

4. Let , and Scalar Multn: Sum or Difference: Product: Quotient: Power: Composition: Properties of Limits Limits also behave as expected when simple operations are performed on them. Because and , It follows that

5. Cancellation & Rationalization Techniques When direct substitution can not be performed because f(c) is undefined, then one of two techniques can be attempted: 1. Cancellation 2. Rationalization These techniques generate a new function that differs only in that the new function is defined at f(c). As a result, direct substitution can be used to find its limit. Functions that differ only at f(c) have the same limit at c. Therefore the limit of the original function is found.

6. Therefore, Note: This is similar to the point discontinuity discussion we had earlier. The simplified function differs from the original rational function only in that it is defined at c. The simplified function has the same limit as the original function and the simplified function’s limit can be found using direct substitution. Cancellation Technique: Example Direct substitution fails because f(-1) causes the denominator to go to 0. Use CANCELLATION to create a function that is defined @ x =c. Factor & cancel f(x): The simplified function differs from the first only at f(c), where x = -1. Functions that differ only at f(c) have the same limit at c.

7. Therefore, Note: A table or graph of the original function can reinforce this conclustion. Rationalization Technique: Example Direct substitution fails because f(0) causes the denominator to go to 0. Use RATIONALIZATION to create a function that is defined @ x=c. Note, that cancellation cannot be used in this case. Rationalize the numerator by multiplying by its conjugate. The new function differs from the first only at f(c), where x = 0. Functions that differ only at f(c) have the same limit at c.

8. – used when f(c) is defined; domain Evaluating Limits Analytically • Direct Substitution • Cancellation Technique • Rationalization Technique • Squeeze Theorem – not covered – used when f(c) is not defined Create a new function by canceling factors or rationalizing the numerator. The new function is defined @ f(c) so that direct substitution can be used to find its limit. Functions that differ only at f(c) have the same limit at c.