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Let's work through the problem step by step.

### The function \( f(x) \) is defined as:

\[
f(x) = \begin{cases} 
3x^3 - 2m, & x \leq -1 \\
4x^2 + 5m, & x > -1 
\end{cases}
\]

#### Part (a) - Find \(\lim_{x \to -1^+} f(x)\) in terms of \(m\):

This limit represents the behavior of the function as \(x\) approaches \(-1\) from the right (i.e., for values greater than \(-1\)).

For \(x > -1\), \( f(x) = 4x^2 + 5m \). So, we evaluate the limit of this expression as \(x \to -1^+\):

\[
\lim_{x \to -1^+} f(x) = 4(-1)^2 + 5m = 4 + 5m
\]

#### Part (b) - Find \(\lim_{x \to -1^-} f(x)\) in terms of \(m\):

This limit represents the behavior of the function as \(x\) approaches \(-1\) from the left (i.e., for values less than or equal to \(-1\)).

For \(x \leq -1\), \( f(x) = 3x^3 - 2m \). So, we evaluate the limit of this expression as \(x \to -1^-\):

\[
\lim_{x \to -1^-} f(x) = 3(-1)^3 - 2m = -3 - 2m
\]

#### Part (c) - Find the value of \(m\) such that:

\[
\lim_{x \to -1^-} f(x) = \lim_{x \to -1^+} f(x)
\]

From parts (a) and (b), we know that:

\[
-3 - 2m = 4 + 5m
\]

Solving this equation for \(m\):

\[
-3 - 2m = 4 + 5m
\]

Add \(2m\) to both sides:

\[
-3 = 4 + 7m
\]

Subtract \(4\) from both sides:

\[
-7 = 7m
\]

Divide by \(7\):

\[
m = -1
\]

### Summary of Results:
- (a) \(\lim_{x \to -1^+} f(x) = 4 + 5m\)
- (b) \(\lim_{x \to -1^-} f(x) = -3 - 2m\)
- (c) The value of \(m\) that makes the limits equal is \(m = -1\).

Would you like further details or explanations on any part of the solution?

### Related Questions:
1. How would the limits change if the function were defined differently for \(x \leq -1\)?
2. What happens to the function's continuity if \(m \neq -1\)?
3. Can you prove the function is continuous at \(x = -1\) for the found value of \(m\)?
4. How do we calculate limits for piecewise functions in general?
5. What is the geometric interpretation of the limits in this context?

### Tip:
Always check both left-hand and right-hand limits to ensure a function is continuous at a point, especially for piecewise functions.

Let f(x) be defined as:
f(x) = { 3x^3 - 2m, x ≤ -1; 4x^2 + 5m, x > -1 }
(a) Find (in terms of m) lim x → -1^+ f(x)
(b) Find (in terms of m) lim x → -1^- f(x)
(c) Find the value of m so that lim x → -1^- f(x) = lim x → -1^+ f(x)

Learn how to evaluate the limit of a piecewise function as x approaches -1 from both directions and find the value of m that ensures continuity. Explore key calculus concepts such as limits, piecewise functions, and continuity in this detailed solution.

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