Sec. 1.3: Evaluating Limits Analytically

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Evaluating Limits Analytically
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Presentation transcript:

Sec. 1.3: Evaluating Limits Analytically The limit of f(x) as x approaches c does not depend on the value of f at c. i.e. The limit of f(x) as x approaches c may not be f(c). Although, for those that are, we could have used direct substitution to evaluate the limit.

Limits Using Direct Substitution If b and c are real numbers and n is a positive integer, then

More Limits Using Direct Substitution If p is a polynomial function, then If r is a rational function r(x) = p(x)/q(x), then

More Limits Using Direct Substitution For radical functions, if n is positive, then the following limit is valid for all c if n is odd, and all c > 0 if n is even.

More Limits Using Direct Substitution For trigonometric functions, if c is in the domain of the function, then

Properties of Limits (Rules) Let b and c be real numbers, let n be a positive integer, and let f and g be functions with the following limits.

More Properties of Limits The limit of a composite function: If f and g are functions such that then

What if direct substitution won’t work? Try the following: If direct substitution gives 0/0, then Factor and use the dividing out strategy. Rationalize and use the dividing out strategy. Simplify and use the dividing out strategy. Still won’t work? Fall back on using a graph or a table.

Sandwich Theorem If f(x)h(x) g(x) for all x in an open interval containing a, except possibly at a itself, and if then exists and is equal to L.

Example Use the Sandwich Theorem to prove that