Limits and Discontinuity Notes
Limits and Discontinuity Notes 201501
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This 4 page Class Notes was uploaded by Hannah Moore on Wednesday February 17, 2016. The Class Notes belongs to 201501 at Clayton State University taught by Giovannitti in Spring 2016. Since its upload, it has received 23 views. For similar materials see Calculus 1 in Math at Clayton State University.
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Date Created: 02/17/16
Limits and Discontinuity Notes Limits of sequences: 1/2 , k approaches ∞, and the fraction approaches 0. 0 is the limit as k approaches ∞. (2 -1)/2 , k approaches ∞, and the fraction approaches 1. 1 is the limit as k approaches ∞. Limits of average rates of change: finding final instantaneous velocity, calculate the rate of change over the last quarter second. Can’t find actual final velocity because can’t use r= ∆s/∆t when ∆t is 0, we just approach a value as ∆t approaches 0(this is a derivative). In general, limits help us discuss what happens when we let things get infinitely lim f (x)=L small, infinitely large, or close to some number. x→ c , as f(x) approaches L, as x gets close to x->c k lim 1/2 =0 k→∞ x-> ∞, x grows without bound. If only matters what happens as x gets closer and closer to c, not what happens lim f (x)≠ f (c) when it actually gets there. x→ c 2 lim f (x)¿2 f(x)=x+1, g(x)=x -1/x-1, both approach x->1, x→1 and f(1)=2. lim g(x)=2 x→1 but g(1) is undefined. f x =¿lim g (x =2 x→1 f(1)≠g(1), lim ¿ x →1 Definition a) Limit: values of f(x) approach L as x approaches c, L is h the limit of f(x) as x approaches c. x→ cf (x)=L b) Limit at infinity: values of f(x) approaches L a x grows without bound, L is the lim f (x)=L limit of f(x) as x approaches ∞. x→ ∞ c) Infinite Limit: values of f(x) grow without bound as x approaches c, then we lim f (x)=∞ say ∞ is the limit of f(x) as x->c. x→ c d) Infinite limit at infinity: values at f(x) grow without bound as x grows without lim f (x)=∞ bound, ∞ is the limit of f(x) as x->∞. c→∞ If limit approaches a real number, the limit exists, but if it approaches ∞ or -∞, the limit does not exist. Definition a) Values of f(x) approach a value L as x approaches c from the left, we say L is x→c−¿ f (x)=L the left-hand limit of f(x) as x approaches c. lim ¿ ¿ b) Values of f(x) approach a value R as x approaches c from the right, R is the x→c+¿ f (x)=R right-hand limit of f(x) as x-c. lim ¿ ¿ x->c- does not mean c is negative or positive. If the two sided limit of f(x) as x->c exists if and only if the left and right limits as x approaches c exist and are equal. Definition Function f has a vertical asymptote at x=c of one or more of the following are true: x→c+¿ f (x)=∞ x→c+¿ f (x)=−∞ x→c−¿ f (x)=−∞ x→c−¿ f (x)=∞,lim ¿ lim ¿ , ¿ , lim ¿ ¿ lim ¿ ¿ ¿ Definition A non-constant function f has a horizontal asymptote at y=L if one or both of the lim f (x)=L lim f (x)=L following are true x→ ∞ , x→−∞ The limit lim f (x)=L means that for all ε> 0, there exist δ>0 such then that if x→ c xε(c-δ, c)U(c, c+δ), them f(x)ε(L-ε,L+ε), f(x) is defined on a punctured interval (c- δ,c)U(c+δ,c) where δ>0 represents a small distance to the left and the right of x=c. c-δ c c+δ Theorem 1.6 Uniqueness of Limits lim f (x)=L lim f (x)=M ,thenL=M a) If x→ c , and x→ c Definition lim f (x)=L Left limit x→ c means for all ε>0, there exists δ>0 such that if xε(c-δ,c) the f(x)ε(L- ε,L+ ε) if and only of |f x −L <| A function is continuous if it’s graph has no breaks, jumps, or holes. Definition lim f (x)= f (c) A function f is continuous at x=c of x→ c x→c−¿ f (x)= f (c) A function f is left continuous a x=c of lim ¿ , and continuous a x=c ¿ x→c+¿ f (x)= f (c) of lim ¿ ¿ A function f is continuous on an interval I if it is continuous at every point in the interval of I. It is right continuous at any closed left endpoint, and left continuous at any closed right endpoint. Definition Continuous on Continuous on Continuous on (1, (1,3) Suppose f is discontinuous at [1 ,3) x=c. 3] We say that x=c is a a) Removable discontinuity of lx→ c (x)butis notequal¿ f (c) Continuous on [,3] x→c+¿ f (x ) x→c−¿ f (x)∧lim ¿ b) Jump discontinuity of ¿ both exist but are not equal lim ¿ ¿ x→c+¿ f (x) x→c−¿ f (x∧lim ¿ c) Infinite discontinuity of one or both of ¿ is infinite. lim ¿ ¿ Theorem 1.16 Continuity of Simple Functions a) Constant, identity, and linear functions are continuous everywhere. In terms of limits, for every K, c, m, and b in R we have lim k=k ,lim x=c,∧lim mx+b=mc+b x→ c x→c x→c b) Power functions are continuous on their domains. In the terms of limits of A is real and K is rational, there for all values x=c at which x is defined we have 2 2 lim A x =Ac x→ c Theorem 1.16 Extreme Value Theorem a) If f is continuous on a closed interval [a,b], then there exists values M and m in the interval [a,b] such that f(M) is the maximum value of f(x) on [a,b] and f(m) is the minimum value of f(x) on [a,b] Theorem 1.18 Intermediate Value Theorem a) If f is continuous on a closed interval [a,b], there for any k strictly between f(a), and f(b) there exists at least one cε(a,b) such then f(c)=k A function can only change signs at rocks and discontinuities. A function can change signs at a point x=c only if f(x) is 0, undefined or discontinuous at x=c