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The question asks to prove the continuity of the function \( f(x) = (x + 2x^3)^4 \) at \( a = -1 \), using the definition of continuity and properties of limits. Here's how we proceed:

### Definition of Continuity:
A function \( f(x) \) is continuous at \( x = a \) if:
1. \( f(a) \) is defined.
2. \( \lim_{x \to a} f(x) \) exists.
3. \( \lim_{x \to a} f(x) = f(a) \).

We will check these three conditions for \( f(x) = (x + 2x^3)^4 \) at \( a = -1 \).

### Step 1: Check if \( f(-1) \) is defined
The function \( f(x) = (x + 2x^3)^4 \) is a polynomial, and polynomials are defined for all real numbers. Let's calculate \( f(-1) \):

\[
f(-1) = \left( -1 + 2(-1)^3 \right)^4 = \left( -1 + 2(-1) \right)^4 = \left( -1 - 2 \right)^4 = (-3)^4 = 81
\]

Thus, \( f(-1) = 81 \), so the first condition is satisfied.

### Step 2: Check if \( \lim_{x \to -1} f(x) \) exists
Since \( f(x) = (x + 2x^3)^4 \) is a polynomial function, and polynomials are continuous everywhere, the limit of \( f(x) \) as \( x \to -1 \) will simply be \( f(-1) \). Thus:

\[
\lim_{x \to -1} f(x) = f(-1) = 81
\]

So, the second condition is satisfied.

### Step 3: Check if \( \lim_{x \to -1} f(x) = f(-1) \)
From the previous two steps, we know that both the limit \( \lim_{x \to -1} f(x) \) and \( f(-1) \) are equal to 81. Thus:

\[
\lim_{x \to -1} f(x) = f(-1) = 81
\]

So, the third condition is satisfied.

### Conclusion:
Since all three conditions of continuity are satisfied, the function \( f(x) = (x + 2x^3)^4 \) is continuous at \( x = -1 \).

Would you like more details or clarifications on any of these steps?

### Related Questions:
1. What are the formal steps to prove continuity using limits?
2. How do we handle discontinuities for non-polynomial functions?
3. What is the difference between continuity and differentiability at a point?
4. How can the concept of limits be applied to non-differentiable points?
5. How can you graphically interpret the continuity of a function?

### Tip:
Polynomials are always continuous and differentiable at all points, which simplifies proving their continuity at specific values.

Use the definition of continuity and the properties of limits to show that the function is continuous at the given number a. f(x) = (x + 2x^3)^4, a = -1

Learn how to prove the continuity of the function f(x) = (x + 2x^3)^4 at x = -1 using the definition of continuity and the properties of limits. This detailed solution walks through the steps to check whether the function is continuous at the given point.

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