Questions: Graph the following function and then find the specified limit. When necessary, state that the limit does not exist. G(x) is defined as: - x^2, for x<2, - x+2, for x>2. Find the limit of G(x) as x approaches 2. Choose the correct graph below. A. B. C. D. Find the limit of G(x) as x approaches 2. Select the correct choice below and fill in any answer boxes in your choice. A. The limit of G(x) as x approaches 2 is . (Type an integer or a simplified fraction.) B. The limit does not exist.

Solution Steps

To find the left-hand limit as x approaches 2, we use the definition of G(x) for x < 2, which is \(G(x) = x^2\). \[\lim_{x \to 2^-} G(x) = \lim_{x \to 2^-} x^2 = 2^2 = 4\]

To find the right-hand limit as x approaches 2, we use the definition of G(x) for x > 2, which is \(G(x) = x + 2\). \[\lim_{x \to 2^+} G(x) = \lim_{x \to 2^+} (x+2) = 2 + 2 = 4\]

Since the left-hand limit and the right-hand limit are both equal to 4, the limit exists and is equal to 4. \[\lim_{x \to 2} G(x) = 4\] The graph that corresponds to the function G(x) is the one where the function approaches 4 from both sides as x approaches 2. The parabola \(y = x^2\) is on the left and the line \(y = x + 2\) is on the right. This is graph A.

Final Answer