evaluating limits
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This document discusses various methods for evaluating limits analytically. It begins by stating that the limit of a function f(x) as x approaches c does not necessarily equal the value of f(c), and that direct substitution can be used to evaluate limits if certain conditions are met. It then provides examples of using direct substitution to evaluate limits of polynomials, rational functions, radical functions, and trigonometric functions as x approaches c. The document also outlines several properties or rules for evaluating limits, such as how limits behave under addition, multiplication, and composition of functions. It notes that if direct substitution does not work, other techniques may be needed to evaluate the limit.
evaluating limits
- 1. Evaluating limits grade 12 Katleho phatoli 201125989
- 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. 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. 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. 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. 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. 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. 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. 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