Calculus Examples

$lim_{x \to - 2} \frac{t^{3} + 8}{t + 2}$

Step 1

Evaluate the limit of $\frac{t^{3} + 8}{t + 2}$ which is constant as $x$ approaches $- 2$ .

$\frac{t^{3} + 8}{t + 2}$

Step 2

Simplify the answer.

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Step 2.1

Simplify the numerator.

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Step 2.1.1

Rewrite $8$ as $2^{3}$ .

$\frac{t^{3} + 2^{3}}{t + 2}$

Step 2.1.2

Since both terms are perfect cubes, factor using the sum of cubes formula, $a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$ where $a = t$ and $b = 2$ .

$\frac{(t + 2) (t^{2} - t \cdot 2 + 2^{2})}{t + 2}$

Step 2.1.3

Simplify.

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Step 2.1.3.1

Multiply $2$ by $- 1$ .

$\frac{(t + 2) (t^{2} - 2 t + 2^{2})}{t + 2}$

Step 2.1.3.2

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

$\frac{(t + 2) (t^{2} - 2 t + 4)}{t + 2}$

Step 2.2

Cancel the common factor of $t + 2$ .

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Step 2.2.1

Cancel the common factor.

$\frac{(t + 2) (t^{2} - 2 t + 4)}{t + 2}$

Step 2.2.2

Divide $t^{2} - 2 t + 4$ by $1$ .

$t^{2} - 2 t + 4$