Suppose g(x) = {x^2−5x if x≤−1
ax^3−7 if x>−1.
Determine a value of the constant a for which lim x→−1 g(x) exists and state the value of the limit, if possible.

Verified step by step guidance

1. Identify the piecewise function g(x) and the point of interest x = -1.
2. For the limit \( \lim_{x \to -1} g(x) \) to exist, the left-hand limit and the right-hand limit as x approaches -1 must be equal.
3. Calculate the left-hand limit: \( \lim_{x \to -1^-} g(x) = \lim_{x \to -1^-} (x^2 - 5x) \).
4. Calculate the right-hand limit: \( \lim_{x \to -1^+} g(x) = \lim_{x \to -1^+} (ax^3 - 7) \).
5. Set the left-hand limit equal to the right-hand limit and solve for the constant a.

Key Concepts

Limit of a Function

The limit of a function describes the value that the function approaches as the input approaches a certain point. In this case, we need to evaluate the limit of g(x) as x approaches -1 from both sides. For the limit to exist, the left-hand limit and the right-hand limit must be equal.

Piecewise Functions

A piecewise function is defined by different expressions based on the input value. In this problem, g(x) has two different expressions depending on whether x is less than or equal to -1 or greater than -1. Understanding how to evaluate each piece is crucial for finding the limit at the point where the definition changes.

Continuity at a Point

A function is continuous at a point if the limit as x approaches that point equals the function's value at that point. For the limit of g(x) to exist at x = -1, we must find a value of 'a' such that the left-hand limit (from x ≤ -1) equals the right-hand limit (from x > -1) and also equals g(-1).