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Calculus, Section 2.5

Learn how to evaluate limits algebraically using substitution and algebraic manipulation techniques. Discover how to handle limits with indeterminate forms and prepare for studying derivatives in Chapter 3. Master important concepts such as factoring polynomials and trigonometric expressions.

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Calculus, Section 2.5

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  1. Calculus, Section 2.5

  2. Evaluating Limits Algebraically Substitution can be used to evaluate limits when the function in question is known to be continuous. For example, is continuous at , and therefore,

  3. Evaluating Limits Algebraically When we study derivatives in Chapter 3, we will be faced with limits where is not defined. In such cases, substitution cannot be used directly. However, many of these limits can be evaluated if we use algebra to rewrite (simplify) the formula for . For example, consider the following limit:

  4. Evaluating Limits Algebraically We say that has an indeterminate form (or is indeterminate) at if the formula for yields an undefined expression of the type When this occurs, our strategy is to transform algebraically, if possible, into a new expression that is defined and continuous at , and then evaluate the limit by substitution (“plugging in”).

  5. Evaluating Limits AlgebraicallyImportant Note It is very important that you be fluent in factoring polynomials and in manipulating trigonometric expressions using the basic trigonometric identities before proceeding in this chapter.

  6. Example 1

  7. Example 2

  8. Example 3

  9. Example 4

  10. Example 5

  11. Example 6

  12. Example 7 In the next example, we will be calculating the limit of a difference quotient as the denominator approaches . This becomes extremely important later on in Chapter 3 in a process known as differentiation.

  13. Example 7

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