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Section 2.3 Techniques for Computing LimitsTopic 1: Limit LawsTheorem Limit LawsAssume limx→afx and limx→agx exist. The following properties hold where c is a real number, and m>0 and n>0 are integers.Sumlimx→afx+gx=limx→afx+limx→agxDifference limx→afx-gx=limx→afx-limx→agx?Constant Multiplelimx→acfx=climx→afxProductlimx→afxgx=limx→afxlimx→agx Quotientlimx→afxgx=limx→afxlimx→agx provided limx→agx≠0Powerlimx→afxn=limx→afxnFractional Powerlimx→afxnm=limx→afxnmprovided f(x)≥0 for x near a if m is even and n/m is reduced to lowest termsTopic 2: Limits of Polynomial and Rational FunctionsTheorem Limits of Polynomial and Rational Functions Assume p and q are polynomials and a is a constant.Polynomial functions:limx→apx=p(a)Rational functions: limx→apxqx=paqa provided q(a)≠0Topic 3: Other Techniques for Finding LimitsDirect substitution cannot be used to find limits when limx→afx exists but limx→afx≠f(a). When this is the case, other strategies must be used to find the limit. Here is a list of some strategies to try when direct substitution does not work.Factor and cancelMultiply by a conjugateSimplify complex fractionsTopic 4: The Squeeze TheoremTheorem The Squeeze Theorem Assume the functions f, g, and h satisfy f(x)≤g(x)≤h(x) for all values of x near a, except possibly at a. If limx→afx=limx→ahx=L, then limx→agx=L. ................
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