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evaluating limits

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evaluating limits

1. 1. Evaluating limits grade 12 Katleho phatoli 201125989
2. 2. 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.
3. 3. Limits Using Direct Substitution If b and c are real numbers and n is a positive integer, then 1.lim 2.lim 3.lim n n x c x c x c b b x c x c
4. 4. 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 lim ( ) ( ) x c p x p c ( ) lim ( ) ( ) ( ) 0 ( )x c p c r x r c q c q c
5. 5. 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. lim n n x c x c
6. 6. More Limits Using Direct Substitution For trigonometric functions, if c is in the domain of the function, then 1. limsin sin 2. limcos cos 3. lim tan tan 4. limcot cot 5. limsec sec 6. limcsc csc x c x c x c x c x c x c x c x c x c x c x c x c
7. 7. 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. lim ( ) lim ( ) 1.Scalar multiple: lim ( ) 2.Sum or difference:lim ( ) ( ) 3. Product: lim ( ) ( ) ( ) 4. Quotient: lim , 0 ( ) 5. Power: lim ( ) x c x c x c x c x c x c n n x c f x L and g x K b f x b L f x g x L K f x g x L K f x L K g x K f x L
8. 8. More Properties of Limits The limit of a composite function: If f and g are functions such that then lim ( ) lim ( ) ( ) x c x L g x L and f x f L lim ( ( )) lim ( ) ( ) x c x c f g x f g x f L
9. 9. What if direct substitution won’t work? In a group 4 of research what happens if direct substation doesn’t work is evaluating a limit. The research is due after a week