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1.5 Infinite Limits & Asymptotes

1.5 Infinite Limits & Asymptotes. After this lesson, you will be able to:. determine infinite limits from the left and from the right find and sketch the vertical asymptotes of the graph of a function. Suppose function.

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1.5 Infinite Limits & Asymptotes

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  1. 1.5 Infinite Limits & Asymptotes

  2. After this lesson, you will be able to: • determine infinite limits from the left and from the right • find and sketch the vertical asymptotes of the graph of a function

  3. Suppose function As the denominator x > 0 gets smaller, the value of the fraction gets larger.

  4. Infinite Limits • If function values keep INCREASING _____________________ as x approaches a given finite value we say the limit is _____________. WITHOUT BOUND INFINITY

  5. Suppose function As the denominator x < 0 gets larger, the value of the fraction gets smaller.

  6. Infinite Limits • If function values keep DECREASING ____________________ as x approaches a given finite value we say the limit is _____________. WITHOUT BOUND - INFINITY

  7. Suppose function

  8. IMPORTANT NOTE: The equal sign in the statement does NOT mean the limit exists! On the contrary, it tells HOW the limit FAILS to exist.

  9. Infinite Limits As the denominator approaches zero, the value of the fraction gets very large. vertical asymptote at x = 0. If the denominator is positive then the fraction is positive. If the denominator is negative then the fraction is negative.

  10. Example 1

  11. Example 1

  12. Definition of a Vertical Asymptote If f (x) approaches infinity or negative infinity as x approaches c from the left or right, then x = c is a vertical asymptote of f. There is a vertical asymptote x = c if: and/or or and/or

  13. Example 2 Determine the limit of the each function as x approaches the given value. 1. 2. 3. 4.

  14. Example 2 Determine all vertical asymptotes of the graph of the each function. 1. 2. 4. 3.

  15. Example 3 Determine all vertical asymptotes of the graph of 1. 2.

  16. Example 4 Find the limit and determine all vertical asymptotes of the graph of the function 1.

  17. Example 4 Find the limit and determine all vertical asymptotes of the graph of the function 2.

  18. Theorem 1.15 Properties of Infinite Limits

  19. Example 5 Determine the limits for all of the functions 1. 2. Because Because and and Property 2 Property 1

  20. Example 5 Determine the limits for all of the functions 3. Because and Property 3 and Property of Limit

  21. Homework Section 1.5: page 85 #1-23 odd, 29-45 odd, 49, 51 (EXCLUDE #17) Try it if you want!

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