Let f(x)={[log(3x)," for "0 = 3]:} Is f continuous at x=3 ? Choose 1 answer: (A) Yes (B) No

Check f(3) is defined: To determine if the function f(x) is continuous at x=3, we need to check if the following conditions are met: 1. f(3) is defined. 2. The limit of f(x) as x approaches 3 from the left (lim x->3^- f(x)) exists. 3. The limit of f(x) as x approaches 3 from the right (lim x->3^+ f(x)) exists. 4. The two one-sided limits are equal to each other and to f(3). First, we will find f(3) using the definition of the function for x >= 3.Find f(3): Substitute x = 3 into the second part of the function definition, (4-x)log(9), to find f(3). f(3) = (4-3)log(9) = log(9) = log(3^2) = 2log(3).Find the limit from the left: Now, we need to find the limit of f(x) as x approaches 3 from the left. This means we will use the first part of the function definition, log(3x), and take the limit as x approaches 3. lim x->3^- f(x) = lim x->3^- log(3x) = log(3*3) = log(9) = 2log(3).Find the limit from the right: Next, we need to find the limit of f(x) as x approaches 3 from the right. This means we will use the second part of the function definition, (4-x)log(9), and take the limit as x approaches 3. lim x->3^+ f(x) = lim x->3^+ (4-x)log(9) = (4-3)log(9) = log(9) = 2log(3).Compare the limits and f(3): We have found that f(3) = 2log(3), the limit from the left is 2log(3), and the limit from the right is also 2log(3). Since all three values are equal, the function f(x) is continuous at x=3.

Resources

Testimonials

Plans

Resources

Testimonials

Plans

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

Let

f(x)={[log(3x)," for "0 < x < 3],[(4-x)log(9)," for "x >= 3]:}
Is
f continuous at
x=3 ?
Choose 1 answer:
(A) Yes
(B) No

Let $\newline$ \( f(x)=\left\{\begin{array}{ll}\log (3 x) & \text { for } 0

View step-by-step help

Home

Math Problems

Calculus

Intermediate Value Theorem

Full solution

Q. Let

\newline

f(x)=\left\{\begin{array}{ll}\log (3 x) & \text { for } 0<x<3 \\ (4-x) \log (9) & \text { for } x \geq 3\end{array}\right.

\newline

f

continuous at

x=3

\newline

Choose

1

answer:

\newline

(A) Yes

\newline

(B) No

Check $f(3)$ is defined: To determine if the function $f(x)$ is continuous at $x=3$ , we need to check if the following conditions are met: $\newline$ $1$ . $f(3)$ is defined. $\newline$ $2$ . The limit of $f(x)$ as $x$ approaches $3$ from the left ( $\lim_{x\to3^-} f(x)$ ) exists. $\newline$ $3$ . The limit of $f(x)$ as $x$ approaches $3$ from the right ( $f(x)$ $1$ ) exists. $\newline$ $4$ . The two one-sided limits are equal to each other and to $f(3)$ . $\newline$ First, we will find $f(3)$ using the definition of the function for $f(x)$ $4$ .
Find $f(3)$ : Substitute $x = 3$ into the second part of the function definition, $(4-x)\log(9)$ , to find $f(3)$ .
$f(3) = (4-3)\log(9) = \log(9) = \log(3^2) = 2\log(3)$ .
Find the limit from the left: Now, we need to find the limit of $f(x)$ as $x$ approaches $3$ from the left. This means we will use the first part of the function definition, $\log(3x)$ , and take the limit as $x$ approaches $3$ . $\lim_{x\to 3^-} f(x) = \lim_{x\to 3^-} \log(3x) = \log(3\cdot 3) = \log(9) = 2\log(3)$ .
Find the limit from the right: Next, we need to find the limit of $f(x)$ as $x$ approaches $3$ from the right. This means we will use the second part of the function definition, $(4-x)\log(9)$ , and take the limit as $x$ approaches $3$ . $\lim_{x\to 3^+} f(x) = \lim_{x\to 3^+} (4-x)\log(9) = (4-3)\log(9) = \log(9) = 2\log(3)$ .
Compare the limits and $f(3)$ : We have found that $f(3) = 2\log(3)$ , the limit from the left is $2\log(3)$ , and the limit from the right is also $2\log(3)$ . Since all three values are equal, the function $f(x)$ is continuous at $x=3$ .