f(x)={[x^(2)," for "x 0]:} Find lim_(x rarr1)f(x). Choose 1 answer: (A) 0 (B) 1 (C) e (D) The limit doesn't exist.

Consider definition of f(x): To find the limit of the piecewise function f(x) as x approaches 1, we need to consider the definition of f(x) for x > 0, since 1 is greater than 0. For x > 0, f(x) is defined as ln(x).Calculate limit of ln(x): We will calculate the limit of ln(x) as x approaches 1 from the right. The limit of ln(x) as x approaches 1 is a standard limit that can be directly evaluated. lim_(x -> 1) ln(x) = ln(1)Evaluate ln(1): We know that the natural logarithm of 1 is 0, because e^0 = 1. ln(1) = 0Determine overall limit of f(x): Since the limit of ln(x) as x approaches 1 from the right is 0, and there is no need to consider the left-hand limit because the function is not defined for x <= 0 at x = 1, the overall limit of f(x) as x approaches 1 is 0.

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Math Problems

Calculus

Find derivatives of logarithmic functions

Full solution

f(x)=\left\{\begin{array}{ll} x^{2} & \text { for } x \leq 0 \\ \ln (x) & \text { for } x>0 \end{array}\right.

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Find

\lim _{x \rightarrow 1} f(x)

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Choose

1

answer:

\newline

(A)

0

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(B)

1

\newline

(C)

e

\newline

(D) The limit doesn't exist.

Consider definition of $f(x)$ : To find the limit of the piecewise function $f(x)$ as $x$ approaches $1$ , we need to consider the definition of $f(x)$ for x > 0, since $1$ is greater than $0$ . For x > 0, $f(x)$ is defined as $f(x)$ $0$ .
Calculate limit of $\ln(x)$ : We will calculate the limit of $\ln(x)$ as $x$ approaches $1$ from the right. The limit of $\ln(x)$ as $x$ approaches $1$ is a standard limit that can be directly evaluated. $\newline$ $\lim_{x \to 1} \ln(x) = \ln(1)$
Evaluate $\ln(1)$ : We know that the natural logarithm of $1$ is $0$ , because $e^0 = 1$ . $\newline$ $\ln(1) = 0$
Determine overall limit of $f(x)$ : Since the limit of $\ln(x)$ as $x$ approaches $1$ from the right is $0$ , and there is no need to consider the left-hand limit because the function is not defined for $x \leq 0$ at $x = 1$ , the overall limit of $f(x)$ as $x$ approaches $1$ is $0$ .