Calculus Examples

$f (x) = {\begin{cases} x^{3}, x \leq 2 \\ a x^{2}, x > 2 \end{cases}$

Step 1

Find the limit of $a x^{2}$ as $x$ approaches $2$ from the right.

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Step 1.1

Change the two-sided limit into a right sided limit.

$lim_{x \to 2^{+}} a x^{2}$

Step 1.2

Evaluate the limit.

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Step 1.2.1

Move the term $a$ outside of the limit because it is constant with respect to $x$ .

$a lim_{x \to 2^{+}} x^{2}$

Step 1.2.2

Move the exponent $2$ from $x^{2}$ outside the limit using the Limits Power Rule.

$a {(lim_{x \to 2^{+}} x)}^{2}$

Step 1.3

Evaluate the limit of $x$ by plugging in $2$ for $x$ .

$a \cdot 2^{2}$

Step 1.4

Simplify the answer.

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Step 1.4.1

Raise $2$ to the power of $2$ .

$a \cdot 4$

Step 1.4.2

Move $4$ to the left of $a$ .

$4 a$

Step 2

Evaluate $x^{3}$ at $2$ .

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Step 2.1

Replace the variable $x$ with $2$ in the expression.

${(2)}^{3}$

Step 2.2

Raise $2$ to the power of $3$ .

$8$

Step 3

Since the limit of $a x^{2}$ as $x$ approaches $2$ from the right is not equal to the function value at $x = 2$ , the function is not continuous at $x = 2$ .

Not continuous

Step 4