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Ch. 2 - Limits and Continuity
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 2 - Limits and ContinuityProblem 4c
Chapter 2, Problem 4c

Which of the following statements about the function y=f(x) graphed here are true, and which are false?


c. limx→1 f(x) does not exist.


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Step 1: Understand the concept of limits. The limit of a function as x approaches a certain value is the value that the function approaches as x gets closer to that value. If the function approaches different values from the left and right, the limit does not exist.
Step 2: Analyze the graph of the function y=f(x) around x=1. Look at the behavior of the function as x approaches 1 from both the left side (x→1⁻) and the right side (x→1⁺).
Step 3: Check if the function approaches the same value from both sides. If the function approaches the same value from both sides, the limit exists. If it approaches different values, the limit does not exist.
Step 4: Consider any discontinuities or jumps in the graph at x=1. If there is a jump or discontinuity, it is likely that the limit does not exist.
Step 5: Conclude whether limx→1 f(x) exists based on your observations of the graph. If the function does not approach the same value from both sides, the statement 'limx→1 f(x) does not exist' is true.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Limits

A limit describes the behavior of a function as its input approaches a certain value. It is essential for understanding continuity and the behavior of functions at specific points. If the function approaches different values from the left and right sides of a point, the limit does not exist at that point.
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One-Sided Limits

Continuity

A function is continuous at a point if the limit as the input approaches that point equals the function's value at that point. If a function has a discontinuity, it may indicate that the limit does not exist. Analyzing continuity helps determine the validity of statements regarding limits.
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Intro to Continuity

Piecewise Functions

Piecewise functions are defined by different expressions based on the input value. Understanding how these functions behave at specific points is crucial for evaluating limits. If the function changes its definition at x=1, it may affect the existence of the limit at that point.
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Piecewise Functions
Related Practice
Textbook Question

Limits and Continuity


Suppose the functions ƒ(x) and g(x) are defined for all x and that lim (x → 0) ƒ(x) = 1/2 and lim (x → 0) g(x) = √2. Find the limits as x → 0 of the following functions.


e. x + ƒ(x)

Textbook Question

Which of the following statements about the function y=f(x) graphed here are true, and which are false?


i. f(0)=1


Textbook Question

Limits and Continuity


Suppose the functions ƒ(x) and g(x) are defined for all x and that lim (x → 0) ƒ(x) = 1/2 and lim (x → 0) g(x) = √2. Find the limits as x → 0 of the following functions.


f. [ƒ(x) • cos x ] / x―1

Textbook Question

Which of the following statements about the function y=f(x) graphed here are true, and which are false?


d. limx→c f(x) exists at every point c in (-1,1).


Textbook Question

Which of the following statements about the function y=f(x) graphed here are true, and which are false?


a. limx→2 f(x) does not exist.


Textbook Question

Which of the following statements about the function y=f(x) graphed here are true, and which are false?


b. limx→2 f(x)=2