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Q. 3

Expert-verified

Calculus

Found in: Page 119

Book edition 1st

Author(s) Peter Kohn, Laura Taalman

Pages 1155 pages

ISBN 9781429241861

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Short Answer

If $f$ is a continuous function, what can you say about $\lim_{x \to 1} f (x) ?$

It can be concluded that there is no hole in the graph of the function at $x = c .$

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1. Given Information.

A continuous function is given.

Step 2. Conclusion.

A function $f$ is continuous at $x = c$ whenever $f (c)$ is defined, $f$ has a limit as $x \to c$ and the value of the limit and the value of the function agree. This guarantees that there is not a hole or jump in the graph of $f$ at $x = c .$

Most popular questions for Math Textbooks

Each function in Exercises 9–12 is discontinuous at some value x = c. Describe the type of discontinuity and any one-sided continuity at x = c, and sketch a possible graph of f.

$\lim_{x \to 2^{-}} f (x) = 2, \lim_{x \to 2^{+}} f (x) = 1, f (2) = 1 .$

Write each of the inequalities in interval notation:

$0 < | x - 2 | < 0.1$

Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(a) A limit exists if there is some real number that it is equal to.

(b) The limit of $f (x)$ as $x \to c$ is the value $f (c)$ .

(c) The limit of $f (x)$ as $x \to c$ might exist even if the value of $f (c)$ does not.

(d) The two-sided limit of $f (x)$ as $x \to c$ exists if and only if the left and right limits of $f (x)$ exists as $x \to c$ .

(e) If the graph of $f$ has a vertical asymptote at $x = 5$ , then $\lim_{x \to 5} f (x) = \infty$ .

(f) If $\lim_{x \to 5} f (x) = \infty$ , then the graph of $f$ has a vertical asymptote at $x = 5$ .

(g) If $\lim_{x \to 2} f (x) = \infty$ , then the graph of $f$ has a horizontal asymptote at $x = 2$ .

(h) If $\lim_{x \to \infty} f (x) = 2$ , then the graph of $f$ has a horizontal asymptote at $y = 2$ .

Calculate each of the limits:

$\lim_{h \to 0} \frac{{(1 + h)}^{3} - 1^{3}}{h}$ .

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$|f (x) - L| < \in$

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