MAT265: Calculus for Engineers IClasswork and Continuity Reference Sheet

31 August, 2015

Name:

Instructions: Complete the following problems with a partner, referring to the referencesheet on the attached page. Please use scratch paper; there is not enough room on this pageto thoroughly show your work. If there is enough time at the end of the class, partners willpresent some of the solutions to these problems on the chalkboard. You do NOT need toturn this assignment in for a grade.

1. (a) From the graph of f below, state the numbers at which f is discontinuous andexplain why.

(b) For each of the numbers stated in part (a), determine whether f is continuous fromthe right, or from the left, or neither.

2. Use the definition of continuity and the properties of limits to show that the functionis continuous at the given number a.

(a) f(x) = 3x4 − 5x + 3√x2 + 4, a = 2

(b) f(x) = (x + 2x3)4, a = −1

3. Use the definition of continuity and the properties of limits to show that the functionf(x) = x

√16− x2 is continuous on the interval [−4, 4].

4. Explain why the function f(x) =

{1

x+2, x 6= −2

1, x = −2is discontinuous at a = −2.

5. Use continuity to evaluate the limit.

(a) limx→45 +√x√

5 + x

(b) limx→π sin(x + sinx)

6. For what value of the constant c is the function f continuous on (−∞,∞)?

f(x) =

{cx2 + 2x, x < 2x3 − cx x ≥ 2.

7. Find the values of a and b that make the function f continuous everywhere.

f(x) =

x2 − 4

x− 2, x < 2

ax2 − bx + 3, 2 ≤ x < 32x− a + b, x ≥ 3.

8. The following functions have removable discontinuities at the given value of a. Find afunction g that agrees with f for x 6= a and is continuous at a.

(a) f(x) =x4 − 1

x− 1, a = 1

(b) f(x) =x3 − x2 − 2x

x− 2, a = 2

The following information can be found in Section 1.5 of your textbook.

Definition: A function f is continuous at a number a if

limx→a

f(x) = f(a).

If f is not continuous at a, we say that f is discontinuous at a .

Definition: A function f is continuous from the right at a number a if

limx→a+

f(x) = f(a)

and f is continuous from the left at a if

limx→a−

f(x) = f(a).

Definition: A function f is continuous on an interval if it is continuous at everynumber in the interval. (If f is defined only on one side of an endpoint of the interval, weunderstand continuous at the endpoint to mean continuous from the right or continuousfrom the left.

Theorem: If f and g are continuous at a and c is a constant, then the following functionsare also continuous at a:

1. f + g

2. f − g

3. cf

4. fg

5. f/g, provided g(a) 6= 0.

Theorem: If f is continuous at b and limx→a g(x) = b, then limx→a f(g(x)) = f(b). Inother words,

limx→a

f(g(x)) = f(

limx→a

g(x)).

Theorem: If g is continuous at a and f is continuous at g(a), then the composite functionf ◦ g given by (f ◦ g)(x) = f(g(x)) is continuous at a.

Theorem: The following types of functions are continuous at every number in theirdomains: polynomials, rational functions, root functions, and trigonometric functions.