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Squeeze Theorem: Proving Limits Analytically

Learn to evaluate limits using the Squeeze Theorem, a valuable method in calculus for proving limits by comparing functions. Explore examples and substitution techniques. Homework exercises provided.

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Squeeze Theorem: Proving Limits Analytically

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  1. 1.3 Evaluating Limits Analytically(Part 2)

  2. Objectives • Evaluate a limit using the Squeeze Theorem.

  3. Squeeze Theorem If f(x) ≤ h(x) ≤ g(x) for all x in an open interval containing c (except possibly at c itself), and if then exists and is equal to L. h is squeezed between g and f.

  4. Squeeze Theorem The Squeeze Theorem is often used to prove other theorems. We can use it to prove Look at figure 1.22 on page 63.

  5. Squeeze Theorem

  6. Using Squeeze Theorem cosθ ≤ sinθ/θ ≤ 1 1≤ sinθ/θ ≤ 1

  7. Example Get 0/0, but you can't factor. So, make it look like sinx/x.

  8. Example sin4x ≠ 4sinx !!! Let y=4x  x=y/4.

  9. Example Let y=3x  x=y/3.

  10. Reminders Let y=7x  x=y/7. Use substitution. If a problem requires substitution, but you don't show any substitution, you will receive NO credit!

  11. Homework 1.3 (page 66) #63-87 odd, #101-104 all (Ignore the directions through #79. Just find the limit.)

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