Show that f is continuous on (−∞, ∞). f(x) = 1 − x2 if x ≤ 1 ln(x) if x > 1 On the interval (−∞, 1), f is ---Select--- a polynomial an exponential a root a logarithmic a rational function; therefore f is continuous on (−∞, 1). On the interval (1, ∞), f is ---Select--- a polynomial an exponential a root a logarithmic a rational function; therefore f is continuous on (1, ∞). At x = 1, lim x→1− f(x) = lim x→1− = , and lim x→1+ f(x) = lim x→1+ = , so lim x→1 f(x) = . Also, f(1) = . Thus, f is continuous at x = 1. We conclude that f is continuous on (−∞, ∞).
Show that f is continuous on (−∞, ∞). f(x) = 1 − x2 if x ≤ 1 ln(x) if x > 1 On the interval (−∞, 1), f is ---Select--- a polynomial an exponential a root a logarithmic a rational function; therefore f is continuous on (−∞, 1). On the interval (1, ∞), f is ---Select--- a polynomial an exponential a root a logarithmic a rational function; therefore f is continuous on (1, ∞). At x = 1, lim x→1− f(x) = lim x→1− = , and lim x→1+ f(x) = lim x→1+ = , so lim x→1 f(x) = . Also, f(1) = . Thus, f is continuous at x = 1. We conclude that f is continuous on (−∞, ∞).
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Question
Show that f is continuous on (−∞, ∞).
f(x) =
| 1 − x2 | if x ≤ 1 |
| ln(x) | if x > 1 |
On the interval
(−∞, 1),
f is ---Select--- a polynomial an exponential a root a logarithmic a rational function; therefore f is continuous on
(−∞, 1).
On the interval
(1, ∞),
f is ---Select--- a polynomial an exponential a root a logarithmic a rational function; therefore f is continuous on
(1, ∞).
At
x = 1,
lim x→1− f(x) = lim x→1−
=
,
and
lim x→1+ f(x) = lim x→1+
=
,
so
lim x→1 f(x) =
.
Also,
f(1) =
.
Thus, f is continuous at
x = 1.
We conclude that f is continuous on
(−∞, ∞).
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