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Published byDaisy McDaniel Modified over 9 years ago
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Finding Limits Analytically 1.3
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Concepts Covered: Properties of Limits Strategies for finding limits The Squeeze Theorem
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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:
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Basic Limits: 1. 2. 3. Examples:
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Properties of Limits: 1. Scalar Multiple: 2. Sum or Difference: 3. Product: 4. Quotient: 5. Power:
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Example using limit properties:
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Find the following:
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Example using limit properties:
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Limits of Trigonometric Functions: You can evaluate the limits of trig functions using direct substitution if. Examples:
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Did you understand? 1. When can you use direct substitution to find a limit?
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If direct substitution will not work…. Try one of these techniques: 1. Cancellation 2. Rationalization
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Cancellation: Try to find a function that agrees with your function at all but. Example:
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Rationalization: Rationalize the numerator.
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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 – x 2 ≤ f(x) ≤ 4 + x 2, find
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Two Special Trig Limits: 1.2. Examples:
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Steps to follow for finding Limits:
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