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$Let h(x)={[(3x)/(xe^(x))," for "x!=0],[k," for "x=0]:} h is continuous for all real numbers. What is the value of k ? Choose 1 answer: (A) 0 (B) 1 (C) 3 (D) e$

Let $h(x)=\left\{\begin{array}{ll}\frac{3 x}{x e^{x}} & \text { for } x \neq 0 \\ k & \text { for } x=0\end{array}\right.$ $\newline$ $h$ is continuous for all real numbers. $\newline$ What is the value of $k$ ? $\newline$ Choose $1$ answer: $\newline$ (A) $0$ $\newline$ (B) $1$ $\newline$ (C) $3$ $\newline$ (D) $e$

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Math Problems

Algebra 1

Domain and range of quadratic functions: equations

Full solution

Q. Let

h(x)=\left\{\begin{array}{ll}\frac{3 x}{x e^{x}} & \text { for } x \neq 0 \\ k & \text { for } x=0\end{array}\right.

\newline

h

is continuous for all real numbers.

\newline

What is the value of

k

\newline

Choose

1

answer:

\newline

(A)

0

\newline

(B)

1

\newline

(C)

3

\newline

(D)

e

Find Limit Right: To determine the value of $k$ that makes $h(x)$ continuous at $x = 0$ , we need to find the limit of $h(x)$ as $x$ approaches $0$ from the left and right and set it equal to $h(0)$ , which is $k$ .
Apply L'Hôpital's Rule: First, let's find the limit of the function as $x$ approaches $0$ from the right ( $x \to 0+$ ). We have the expression $\frac{3x}{xe^{x}}$ . As $x$ approaches $0$ , the numerator approaches $0$ and the denominator approaches $0$ as well, since $e^{0} = 1$ . We can apply L'Hôpital's Rule because we have an indeterminate form of $0/0$ .
Differentiate Numerator and Denominator: Applying L'Hôpital's Rule, we differentiate the numerator and the denominator with respect to $x$ . The derivative of the numerator, $3x$ , with respect to $x$ is $3$ . The derivative of the denominator, $x*e^x$ , with respect to $x$ is $e^x + x*e^x$ using the product rule.
Take Limit Again: Now, we take the limit of the new function as $x$ approaches $0$ . The limit of $\frac{3}{e^x + x*e^x}$ as $x$ approaches $0$ is $\frac{3}{1 + 0} = 3$ .
Determine Value of $k$ : Since $h(x)$ is continuous for all real numbers, the limit as $x$ approaches $0$ from the right must equal $h(0)$ , which is $k$ . Therefore, $k$ must be equal to $3$ .