Calculus Examples

$lim_{z \to - 1} \frac{3 x^{2} + x + 1}{x^{3} + 1}$

Step 1

Evaluate the limit of $\frac{3 x^{2} + x + 1}{x^{3} + 1}$ which is constant as $z$ approaches $- 1$ .

$\frac{3 x^{2} + x + 1}{x^{3} + 1}$

Step 2

Simplify the denominator.

Tap for more steps...

Step 2.1

Rewrite $1$ as $1^{3}$ .

$\frac{3 x^{2} + x + 1}{x^{3} + 1^{3}}$

Step 2.2

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 $a = x$ and $b = 1$ .

$\frac{3 x^{2} + x + 1}{(x + 1) (x^{2} - x \cdot 1 + 1^{2})}$

Step 2.3

Simplify.

Tap for more steps...

Step 2.3.1

Multiply $- 1$ by $1$ .

$\frac{3 x^{2} + x + 1}{(x + 1) (x^{2} - x + 1^{2})}$

Step 2.3.2

One to any power is one.

$\frac{3 x^{2} + x + 1}{(x + 1) (x^{2} - x + 1)}$