g(x)={[ln(x)," for "0

Question

g(x)={[ln(x)," for "0 < x <= 2],[x^(2)ln(2)," for "x > 2]:} Find  lim_(x rarr2)g(x). Choose 1 answer: (A)  ln(2) (B) 4 (C)  4*ln(2) (D) The limit doesn't exist.

Bytelearn · Accepted Answer

Define Function: We need to find the limit of the function g(x) as x approaches 2. The function g(x) is defined differently on two intervals: for $0 < x \leq 2$, $g(x) = \ln(x)$, and for $x > 2$, $g(x) = x^2\ln(2)$. To find the limit as $x$ approaches 2, we need to consider the value of the function as $x$ approaches 2 from the left and from the right.Limit from Left: First, let's find the limit as $x$ approaches 2 from the left, which means we are looking at the interval $0 < x \leq 2$. In this interval, the function is defined as $g(x) = \ln(x)$. So, we need to evaluate $\lim_{x \to 2^-} \ln(x)$.Limit from Right: Since the natural logarithm function, $\ln(x)$, is continuous for $x > 0$, we can directly substitute the value of $x$ with 2 to find the limit from the left. Therefore, $\lim_{x \to 2^-} \ln(x) = \ln(2)$.Check Equality: Now, let's find the limit as $x$ approaches 2 from the right, which means we are looking at the interval $x > 2$. In this interval, the function is defined as $g(x) = x^2\ln(2)$. So, we need to evaluate $\lim_{x \to 2^+} x^2\ln(2)$.Conclusion: Since the function $x^2\ln(2)$ is a product of a constant, $\ln(2)$, and a polynomial, $x^2$, which are both continuous everywhere, we can directly substitute the value of $x$ with 2 to find the limit from the right. Therefore, $\lim_{x \to 2^+} x^2\ln(2) = 2^2\ln(2) = 4\ln(2)$.To determine if the limit of $g(x)$ as $x$ approaches 2 exists, we need to check if the limits from the left and the right are equal. We found that $\lim_{x \to 2^-} \ln(x) = \ln(2)$ and $\lim_{x \to 2^+} x^2\ln(2) = 4\ln(2)$. Since $\ln(2) \neq 4\ln(2)$, the limits from the left and the right are not equal.Because the limits from the left and the right are not equal, the limit of $g(x)$ as $x$ approaches 2 does not exist. Therefore, the correct answer is (D) The limit doesn't exist.

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