Chapter 2 | Limits
211
• You can evaluate the limit of a function by factoring and canceling, by multiplying by a conjugate, or by simplifying a complex fraction. • The squeeze theorem allows you to find the limit of a function if the function is always greater than one function and less than another function with limits that are known. 2.4 Continuity • For a function to be continuous at a point, it must be defined at that point, its limit must exist at the point, and the value of the function at that point must equal the value of the limit at that point. • Discontinuities may be classified as removable, jump, or infinite. • A function is continuous over an open interval if it is continuous at every point in the interval. It is continuous over a closed interval if it is continuous at every point in its interior and is continuous at its endpoints. • The composite function theorem states: If f ( x ) is continuous at L and lim x → a g ( x ) = L , then lim x → a f ⎛ ⎝ g ( x ) ⎞ ⎠ = f ⎛ ⎝ lim x → a g ( x ) ⎞ ⎠ = f ( L ). • The Intermediate Value Theorem guarantees that if a function is continuous over a closed interval, then the function takes on every value between the values at its endpoints. 2.5 The Precise Definition of a Limit • The intuitive notion of a limit may be converted into a rigorous mathematical definition known as the epsilon-delta definition of the limit . • The epsilon-delta definition may be used to prove statements about limits. • The epsilon-delta definition of a limit may be modified to define one-sided limits.
CHAPTER 2 REVIEW EXERCISES True or False . In the following exercises, justify your answer with a proof or a counterexample. 208. A function has to be continuous at x = a if the lim x → a f ( x ) exists.
212. Using the graph, find each limit or explain why the limit does not exist. a. lim x →−1 f ( x ) b. lim x →1 f ( x ) c. lim x →0 + f ( x ) d. lim x →2 f ( x )
209. You can use the quotient rule to evaluate lim x →0 sin x x .
210. If there is a vertical asymptote at x = a for the function f ( x ), then f is undefined at the point x = a .
f ( x ) does not exist, then f is undefined at the
211. If lim x → a point x = a .
In the following exercises, evaluate the limit algebraically or explain why the limit does not exist. 213. lim x →2 2 x 2 −3 x −2 x −2
Made with FlippingBook - professional solution for displaying marketing and sales documents online