### Create a StudySoup account

#### Be part of our community, it's free to join!

Already have a StudySoup account? Login here

# Limits and Discontinuity Notes 201501

Clayton State

### View Full Document

## About this Document

## 23

## 0

## Popular in Calculus 1

## Popular in Math

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.

## Similar to 201501 at Clayton State

## Reviews for Limits and Discontinuity Notes

### What is Karma?

#### Karma is the currency of StudySoup.

#### You can buy or earn more Karma at anytime and redeem it for class notes, study guides, flashcards, and more!

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

### BOOM! Enjoy Your Free Notes!

We've added these Notes to your profile, click here to view them now.

### You're already Subscribed!

Looks like you've already subscribed to StudySoup, you won't need to purchase another subscription to get this material. To access this material simply click 'View Full Document'

## Why people love StudySoup

#### "I was shooting for a perfect 4.0 GPA this semester. Having StudySoup as a study aid was critical to helping me achieve my goal...and I nailed it!"

#### "Selling my MCAT study guides and notes has been a great source of side revenue while I'm in school. Some months I'm making over $500! Plus, it makes me happy knowing that I'm helping future med students with their MCAT."

#### "Knowing I can count on the Elite Notetaker in my class allows me to focus on what the professor is saying instead of just scribbling notes the whole time and falling behind."

#### "It's a great way for students to improve their educational experience and it seemed like a product that everybody wants, so all the people participating are winning."

### Refund Policy

#### STUDYSOUP CANCELLATION POLICY

All subscriptions to StudySoup are paid in full at the time of subscribing. To change your credit card information or to cancel your subscription, go to "Edit Settings". All credit card information will be available there. If you should decide to cancel your subscription, it will continue to be valid until the next payment period, as all payments for the current period were made in advance. For special circumstances, please email support@studysoup.com

#### STUDYSOUP REFUND POLICY

StudySoup has more than 1 million course-specific study resources to help students study smarter. If you’re having trouble finding what you’re looking for, our customer support team can help you find what you need! Feel free to contact them here: support@studysoup.com

Recurring Subscriptions: If you have canceled your recurring subscription on the day of renewal and have not downloaded any documents, you may request a refund by submitting an email to support@studysoup.com

Satisfaction Guarantee: If you’re not satisfied with your subscription, you can contact us for further help. Contact must be made within 3 business days of your subscription purchase and your refund request will be subject for review.

Please Note: Refunds can never be provided more than 30 days after the initial purchase date regardless of your activity on the site.