Problem 1 Verify that \(f(x)=3 x+4\) is co... [FREE SOLUTION]

Chapter 1: Problem 1

Verify that \(f(x)=3 x+4\) is continuous at \(x=5\).

Short Answer

Expert verified

Yes, \(f(x)\) is continuous at \(x=5\) because \(f(5)=19\) and the limit \(\lim_{{x \to 5}} f(x) = 19\).

Step by step solution

Define Continuity

A function is continuous at a point if three conditions are met: the function is defined at the point, the limit exists at that point, and the value of the function at that point equals the limit.

Check if f(5) is Defined

We need to check if the function is defined at \(x=5\). Substitute \(x=5\) into \(f(x)\): \[ f(5) = 3(5) + 4 = 15 + 4 = 19 \] Thus, \(f(5)\) is defined and equals 19.

Calculate the Limit as x Approaches 5

Calculate the limit of \(f(x)\) as \(x\) approaches 5: \[ \lim_{{x \to 5}} (3x+4) \] Substitute \(x=5\): \[ \lim_{{x \to 5}} (3(5) + 4) = 19 \] The limit exists and equals 19.

Compare Limit and Function Value

Verify that the limit as \(x\) approaches 5 is equal to \(f(5)\). We have: \(\lim_{{x \to 5}} f(x) = 19\) and \(f(5) = 19\). Since both are equal, \(f(x)\) is continuous at \(x=5\).

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Limit

Limits are fundamental in understanding how functions behave as the input approaches a particular value. A limit essentially tells us what value a function approaches as the input approaches some point. This is important for assessing continuity. To illustrate, consider a function like \( f(x) = 3x + 4 \). As \( x \) gets closer and closer to 5, the value of \( f(x) \) approaches the number we find by substituting 5 into the function. Therefore,

For \( f(x) = 3x + 4 \), the calculation is straightforward: \( \lim_{{x \to 5}} f(x) = 3(5) + 4 \).
This turns into \( \lim_{{x \to 5}} f(x) = 15 + 4 = 19 \).

Suppose the output approaches the same number back from the right and left, then we say the limit as \( x \to 5 \) exists. Limits, therefore, are essential tools that help gauge the behavior of functions near specific points.

Function Definition

Understanding the definition of a function at a certain point is critical when discussing continuity. A function is defined at a point if it assigns exactly one value to its input at that point. Consider \( f(x) = 3x + 4 \). This linear function will exist for all real \( x \), including 5. To verify this:

We substituted \( x = 5 \) into the function: \( f(5) = 3(5) + 4 \).
Solving gives us \( f(5) = 15 + 4 = 19 \).

Thus, the function is defined at \( x = 5 \) because we can calculate a specific value, which is 19 in this instance. A clear function definition at the point indicates there are no breaks or holes here, contributing to continuity.

Evaluating Limits

Evaluating limits at a specific point is crucial in determining continuity. We do so by ensuring that our calculated limit value matches the actual output of the function at that point. For \( f(x) = 3x + 4 \), we need to evaluate the limit as \( x \to 5 \) and check against \( f(5) \):

First, calculate the limit: \( \lim_{{x \to 5}} (3x + 4) = 19 \).
Then, we evaluated the function: \( f(5) = 19 \).

Both the limit and the function value are the same, which confirms continuity at \( x = 5 \). Evaluating limits essentially ensure that no unexpected jumps or yawning gaps occur at the point in question. This tool confirms that approaching from either side results in convergent and expected values.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Verify that \(f(x)=3 x+4\) is continuous at \(x=5\).

Short Answer

Step by step solution

Define Continuity

Check if f(5) is Defined

Calculate the Limit as x Approaches 5

Compare Limit and Function Value

Key Concepts

Limit

Function Definition

Evaluating Limits

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Discrete Mathematics

Statistics

Mechanics Maths

Pure Maths

Applied Mathematics

Study anywhere. Anytime. Across all devices.