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Finding Limits Analytically
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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.
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Basic Limits: Examples:
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Properties of Limits: Scalar Multiple: Sum or Difference: Product:
Quotient: Power:
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Example using limit properties:
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Example using limit properties:
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? 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: Cancellation 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 – x2 ≤ f(x) ≤ 4 + x2, find
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Two Special Trig Limits:
Examples:
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Steps to follow for finding Limits:
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