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Mastering Analytical Limit Finding Strategies

Explore properties of limits, techniques to find limits, and concepts like the Squeeze Theorem. Practice examples included. Learn limit properties, trigonometric functions, and techniques for challenging cases. Discover direct substitution and alternative methods for finding limits.

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Mastering Analytical Limit Finding Strategies

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  1. Finding Limits Analytically 1.3

  2. Concepts Covered: • Properties of Limits • Strategies for finding limits • The Squeeze Theorem

  3. From section 1.2: The limit of f(x) as x approaches c does not depend on the value of f at x = c. • Example: • However, sometimes the limit may be exactly f(c). When this occurs the function is at x = c. In this case find the limit by substitution. • Example:

  4. Basic Limits: 1. 2. 3. Examples:

  5. Properties of Limits: • Scalar Multiple: • Sum or Difference: • Product: • Quotient: • Power:

  6. Example using limit properties:

  7. Example using limit properties: Find the following:

  8. Example using limit properties:

  9. Limits of Trigonometric Functions: You can evaluate the limits of trig functions using direct substitution if . Examples:

  10. Did you understand? • When can you use direct substitution to find a limit?

  11. If direct substitution will not work…. Try one of these techniques: • Cancellation • Rationalization

  12. Cancellation: Try to find a function that agrees with your function at all but . Example:

  13. Rationalization: Rationalize the numerator.

  14. The Squeeze Theorem: • If h(x) ≤ f(x) ≤ g(x) for all x in an open interval containing c, except for possibly at c itself and if , then • Example: If 4 – x2 ≤ f(x) ≤ 4 + x2, find

  15. Two Special Trig Limits: 1. 2. Examples:

  16. Steps to follow for finding Limits:

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