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For the function "},{"type":"inlineMath","attrs":{"latex":"g(x) = \\begin{cases} x^2 & x < 0 \\\\ e^x - 1 & x > 0 \\end{cases}"}},{"type":"text","marks":[{"type":"bold"}],"text":", does "},{"type":"inlineMath","attrs":{"latex":"\\displaystyle \\lim_{x \\to 0} g(x)"}},{"type":"text","marks":[{"type":"bold"}],"text":" exist? If so, find it. If not, explain."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Limits and Piecewise Functions"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to evaluate one-sided limits for a piecewise function and determine if the overall limit exists."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Left-hand limit: "},{"type":"inlineMath","attrs":{"latex":"\\displaystyle \\lim_{x \\to 0^-} g(x)"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Right-hand limit: "},{"type":"inlineMath","attrs":{"latex":"\\displaystyle \\lim_{x \\to 0^+} g(x)"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"The limit exists if both one-sided limits are equal."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Find the left-hand limit as "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" approaches $0"},{"type":"inlineMath","attrs":{"latex":" from below using "}},{"type":"text","text":"g(x) = x^2$."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Find the right-hand limi