We are tasked with evaluating the limits of the derivative of the piecewise function $f(x)$ as $x$ approaches 1 from the left ($x \to 1^-$) and from the right ($x \to 1^+$). We will also determine whether $f(x)$ is differentiable at $x = 1$.

Step 1: Evaluate $\lim_{x \to 1^-} f'(x)$

For $x < 1$, the function is given by $f(x) = x^2$.

The derivative of $f(x) = x^2$ is: $f'(x) = 2x$ Thus, as $x \to 1^-$: $\lim_{x \to 1^-} f'(x) = \lim_{x \to 1^-} 2x = 2(1) = 2$

Step 2: Evaluate $\lim_{x \to 1^+} f'(x)$

For $1 \leq x < 3$, the function is given by $f(x) = 2x + 1$.

The derivative of $f(x) = 2x + 1$ is: $f'(x) = 2$ Thus, as $x \to 1^+$: $\lim_{x \to 1^+} f'(x) = \lim_{x \to 1^+} 2 = 2$

Step 3: Determine whether $f(x)$ is differentiable at $x = 1$

To determine if $f(x)$ is differentiable at $x = 1$, the following conditions must hold:

1. $f(x)$ must be continuous at $x = 1$.
2. The derivative from the left must equal the derivative from the right, i.e., $\lim_{x \to 1^-} f'(x) = \lim_{x \to 1^+} f'(x)$.

Continuity at $x = 1$:

We check the value of $f(x)$ at $x = 1$. From the piecewise definition:

• For $x < 1$, $f(x) = x^2$, so $f(1^-) = 1^2 = 1$.
• For $x \geq 1$, $f(x) = 2x + 1$, so $f(1^+) = 2(1) + 1 = 3$.

Since $f(1^-) = 1$ and $f(1^+) = 3$, the function $f(x)$ is not continuous at $x = 1$.

Derivatives at $x = 1$:

We already found: $\lim_{x \to 1^-} f'(x) = 2 \quad \text{and} \quad \lim_{x \to 1^+} f'(x) = 2$ Since these two limits are equal, the derivative from the left and right matches.

Conclusion:

Even though the derivative from the left and right is the same ($f'(x) = 2$ as $x \to 1$), the function is not continuous at $x = 1$. A function must be continuous at a point to be differentiable there. Hence, $f(x)$ is not differentiable at $x = 1$ because it is not continuous at that point.

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Would you like further details on any step or have any questions? Here are some related questions to explore:

1. What is the formal definition of differentiability at a point?
2. How does continuity relate to differentiability?
3. Can a function have a continuous derivative but still be non-differentiable at a point?
4. What are other conditions that can prevent differentiability at a point?
5. How does the piecewise nature of a function affect its differentiability?

Tip: Always check both continuity and the equality of derivatives from both sides when verifying differentiability at a point.