If f(x)=x^3 then prove it is continuous at x=1

To prove that \( f(x) = x^3 \) is continuous at \( x = 1 \), we follow the definition of continuity at a point:

A function \( f(x) \) is continuous at \( x = c \) if:
1. \( f(c) \) is defined.
2. The limit \( \lim_{x \to c} f(x) \) exists.
3. \( \lim_{x \to c} f(x) = f(c) \).

### Step 1: Verify \( f(1) \) is defined.
For \( f(x) = x^3 \),
\[
f(1) = 1^3 = 1.
\]
Thus, \( f(1) \) is defined.

### Step 2: Find \( \lim_{x \to 1} f(x) \).
To compute the limit:
\[
\lim_{x \to 1} f(x) = \lim_{x \to 1} x^3.
\]
Since \( x^3 \) is a polynomial, it is continuous everywhere, so:
\[
\lim_{x \to 1} x^3 = 1^3 = 1.
\]

### Step 3: Check if \( \lim_{x \to 1} f(x) = f(1) \).
We have:
\[
\lim_{x \to 1} f(x) = 1, \quad f(1) = 1.
\]
Since \( \lim_{x \to 1} f(x) = f(1) \), \( f(x) = x^3 \) is continuous at \( x = 1 \).

### Conclusion:
Thus, \( f(x) = x^3 \) is continuous at \( x = 1 \).

Would you like a deeper dive into any part of this proof?

Here are 5 related questions for exploration:
1. How can we extend this proof to show continuity at any point \( x = c \)?
2. What is the graphical interpretation of continuity for \( f(x) = x^3 \)?
3. How does the continuity of \( x^3 \) differ from that of \( |x| \) at \( x = 0 \)?
4. Why are polynomials always continuous?
5. How does the definition of continuity relate to differentiability?

**Tip**: Always check limits from both sides for piecewise functions to ensure continuity!

Understand how to prove the continuity of the function f(x) = x^3 at x = 1 using limit laws and the formal definition of continuity. This step-by-step solution covers essential calculus concepts and is designed for students in Grades 11-12.

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