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  1. Title HW pp. 58: DO NOW! Estimate numerically: 1) 2) • Objective • Solve limits analytically by: • Direct substitution; • Dividing out; and • Rationalizing the numerator. Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Vivamus et magna. Fusce sed sem sed magna suscipit egestas. Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Vivamus et magna. Fusce sed sem sed magna suscipit egestas.

  2. Solving Limits Analytically Limit Laws, Direct Substitution, Dividing out, and rationalizing the numerator

  3. Methods of Finding Limits • Numerically = in a table • Graphically = on a graph • Analytically = using algebra/equations

  4. Solving Limits Analytically • Limit Laws • Direct Substitution • Dividing Out • Rationalizing the Numerator

  5. Limit Laws Let b, c , n   > 0, and Then:

  6. Limit Laws Let b, c , n   > 0, and Then: 4. 5.

  7. Direct Substitution This method of solution can only be used when the functions are: • Polynomial functions • Rational functions with nonzero denominators • Positive Integer Radical functions (square, cube, quartic, etc roots) • Trigonometric functions • Composite functions of above

  8. Example Find the limit for each function.

  9. When the limit gives an indeterminant form…

  10. Dividing Out When you have a rational function where you can “divide out” a factor.

  11. Rationalizing the Numerator When you have a rational function where the numerator has a sum or differenceand a radical.

  12. Classwork Find the limit of each function, analytically.

  13. Title HW pp. 66: 67-80 all DO NOW! Find each limit: 1) 2) • Objective • Solve limits analytically by: • Simplifying • Direct Substitution • Rationalizing the Numerator • Trig rules Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Vivamus et magna. Fusce sed sem sed magna suscipit egestas. Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Vivamus et magna. Fusce sed sem sed magna suscipit egestas.

  14. Trig Limits Sometimes the limit of a trig function is indeterminant.In this case, remember these:

  15. Trig Limits Use limit laws, trig props, and old concepts from algebra. 1.) 2.)

  16. Trig Limits Use limit laws, trig props, and old concepts from algebra. 3.) 4.)

  17. Trig Limits Use limit laws, trig props, and old concepts from algebra. 5.) 6.)

  18. Trig Limits Use limit laws, trig props, and old concepts from algebra. 7.)

  19. Classwork Complete the joke worksheets provided. HW: p. 66 from worksheet: 67-80 all.