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Finding Limits Analytically: Direct Substitution and Other Methods

Learn how to find limits analytically using direct substitution and other methods such as factoring, rationalizing, and eliminating embedded denominators. Discover properties of limits and explore examples to understand how to evaluate limits analytically.

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Finding Limits Analytically: Direct Substitution and Other Methods

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  1. Bell Ringer

  2. Bell Ringer continued… Does the limit exist at x = 1 if so what is it?

  3. Finding Limits Analytically

  4. Strategies for Finding Limits To find limits analytically, try the following: • Direct Substitution (Always 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

  5. 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.

  6. 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:

  7. 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:

  8. Example Let and . Find the following limits.

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

  10. 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:

  11. 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:

  12. Example 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:

  13. Indeterminate Form 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… Evaluate the limit analytically: An example of an indeterminate form because the limit can not be determined. 1/0 is another example. How can we simplify: ?

  14. REVIEW 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

  15. Example 1 Evaluate the limit analytically: Factor the numerator and denominator At first Direct Substitution fails because x=2 results in dividing by zero Cancel common factors This function is equivalent to the original function except at x=2 Direct substitution

  16. Example 2 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

  17. Example 3 Evaluate the limit analytically: Expand the the expression to see if anything cancels Factor to see if anything cancels Direct substitution

  18. Example 4 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

  19. Example 5 Evaluate the limit analytically: Rationalize the numerator by multiplying by conjugate Cancel common factors Direct substitution

  20. The Sandwich Theorem

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