Problem 36 In Problems $34-37$, is the fu... [FREE SOLUTION]

91影视

Calculus

Deborah Hughes-Hallett, William G. McCallum, Andrew M. Gleason

$Math Studyset 91影视 Explanations$ Math

6 Edition

Chapter 1: Problem 36

In Problems $34-37$, is the function continuous for all $x ?$ If not, say where it is not continuous and explain in what way the definition of continuity is not satisfied. $$f(x)=\left\\{\begin{array}{ll} x / x & x \neq 0 \\ 1 & x=0 \end{array}\right.$$

Short Answer

Expert verified

The function is continuous for all $ x $.

Step by step solution

Review Function Definition

The function is defined as $ f(x) = \frac{x}{x} $ for $ x eq 0 $ and $ f(x) = 1 $ for $ x = 0 $. This can be simplified as $ f(x) = 1 $ for all $ x eq 0 $, since $ \frac{x}{x} = 1 $ when $ x eq 0 $. Thus, $ f(x) = 1 $ for all $ x $, but with an explicit definition at $ x = 0 $.

Check Continuity Definition

A function $ f(x) $ is continuous at a point $ x = c $ if the following conditions hold: 1) $ f(c) $ is defined, 2) $ \lim_{x \to c} f(x) $ exists, and 3) $ \lim_{x \to c} f(x) = f(c) $. We need to check these conditions at $ x = 0 $, the only point where definition changes.

Evaluate Limit as x Approaches 0

For $ x eq 0 $, $ f(x) = 1 $. Thus, the limit as $ x \to 0 $ is $ \lim_{x \to 0} f(x) = 1 $.

Verify the Function Value at x = 0

The function value directly given at $ x = 0 $ is $ f(0) = 1 $.

Compare Limit and Function Value

Since $ \lim_{x \to 0} f(x) = 1 $ and $ f(0) = 1 $, the function is continuous at $ x = 0 $ because the limit and the function value agree.

Assess Continuity for Other Values of x

For all other $ x eq 0 $, $ f(x) $ is constantly $ 1 $, meaning it is continuous everywhere else by nature of a constant function.

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Key Concepts

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

Piecewise Functions

A piecewise function is an interesting type of mathematical function. It is defined by different expressions based on different intervals of the independent variable. Imagine a cake cut into sections, each with its own unique flavor or icing! Each piece is delicious on its own, but together they make a complete cake.

For example, in the function provided in our exercise, we have:

For all values of $x$ not equal to zero, the function is defined as $f(x) = \frac{x}{x} = 1$. This is because any number divided by itself (except zero) is 1.
At the point $x = 0$, the function is specifically defined as $f(x) = 1$.

The idea here is that the function adjusts like a chameleon based on the value of $x$, but it remains well-behaved and continuous, sticking to the value 1. The explicit definition at $x = 0$ ensures the function is consistent across all inputs.

Limit of a Function

The limit of a function is an essential concept in calculus. It helps us understand the behavior of functions as they approach a particular point. You can think of trying to understand someone's mood by observing their expressions as they come closer to you!

In mathematical terms, the limit tells us what value a function approaches as the input gets very close to a specific number. For our piecewise function example:

The limit as $x$ approaches zero is calculated by observing the value that $f(x)$ approaches as $x$ gets closer and closer to zero, from both directions (left and right).
Since $f(x) = 1$ for all $x eq 0$, the limit is quite simple: $\lim_{x \to 0} f(x) = 1$.

This consistent behavior of the function around $x = 0$ guarantees that the limit exists, and supports our journey to verify continuity.

Definition of Continuity

Continuity in mathematics describes how smooth a function graph is. If you can trace the graph of a function with a pen without lifting it, the function is continuous. It's like walking on a straight, seamless path without any breaks or jumps!

Mathematically, a function $f(x)$ is considered continuous at a point $x = c$ if:

$f(c)$ is defined.
The limit $\lim_{x \to c} f(x)$ exists.
The limit at that point equals the function's value, meaning $\lim_{x \to c} f(x) = f(c)$.

In our function, let's inspect the continuity at $x = 0$:

$f(0) = 1$ is defined.
The limit $\lim_{x \to 0} f(x) = 1$ exists.
Both the limit and the function value at $x = 0$ match: $\lim_{x \to 0} f(x) = f(0)$.

Hence, the function is continuous at $x = 0$. For all other $x eq 0$, the function is always 1, maintaining continuity. Thus, our piecewise function, though defined differently over intervals, manages to keep a smooth progression throughout.

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91影视

In Problems \(34-37\), is the function continuous for all \(x ?\) If not, say where it is not continuous and explain in what way the definition of continuity is not satisfied. $$f(x)=\left\\{\begin{array}{ll} x / x & x \neq 0 \\ 1 & x=0 \end{array}\right.$$

Short Answer

Step by step solution

Review Function Definition

Check Continuity Definition

Evaluate Limit as x Approaches 0

Verify the Function Value at x = 0

Compare Limit and Function Value

Assess Continuity for Other Values of x

Key Concepts

Piecewise Functions

Limit of a Function

Definition of Continuity

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