Tuesday, December 11, 2018

Single Variable Calculus, Chapter 2, Review Exercises, Section Review Exercises, Problem 23

Suppose that

f(x)={x if x<03x if 0x<3(x3)2 if x>3

a.) Find each limit. if it exists

i.)lim

\qquad \qquad Using Equation 2, \lim \limits_{x \to 0^+} f(x) = \lim \limits_{x \to 0^+} (3 - x) = 3-0 = 3

\qquad ii.) \lim \limits_{x \to 0^-} f(x)

\qquad \qquad Using Equation 1, \lim \limits_{x \to 0^-} f(x) = \lim \limits_{x \to 0^-} \sqrt{-x} = \sqrt{0} = 0

\qquad iii.) \lim \limits_{x \to 0} f(x)

\qquad \qquad Referring to the given conditions, \lim \limits_{x \to 0} f(x) does not exist

\qquad iv.) \lim \limits_{x \to 3^-} f(x)

\qquad \qquad Using Equation 2, \lim \limits_{x \to 3^-} f(x) = \lim \limits_{x \to 3^-} (3 -x) = 3 - 3 = 0

\qquad v.) \lim \limits_{x \to 3^+} f(x)

\qquad \qquad Using Equation 3, \lim \limits_{x \to 3^+} f(x) = \lim \limits_{x \to 3^+} (x - 3)^2 = (3 - 3)^2 = 0

\qquad vi.) \lim \limits_{x \to 3} f(x)

\qquad \qquad Referring to the given conditions, \lim \limits_{x \to 3} f(x) = 0

b.) Find where f is discontinuous.

\qquad Referring to the given conditions f is discontinuous at x = 0 because \lim \limits_{x \to 0} f(x) does not exist and also f is discontinuous at x = 3 because f(3) is undefined.

\qquad Therefore f is discontinuous at 0 and 3.

c.) Graph the function f.

No comments:

Post a Comment

Why is the fact that the Americans are helping the Russians important?

In the late author Tom Clancy’s first novel, The Hunt for Red October, the assistance rendered to the Russians by the United States is impor...