Precalculus Examples

$\frac{lim_{x \to 2} x^{4} + x^{3} - 24}{x^{2} - 4}$

Step 1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $2$ .

$\frac{lim_{x \to 2} x^{4} + lim_{x \to 2} x^{3} - lim_{x \to 2} 24}{x^{2} - 4}$

Step 2

Move the exponent $4$ from $x^{4}$ outside the limit using the Limits Power Rule.

$\frac{{(lim_{x \to 2} x)}^{4} + lim_{x \to 2} x^{3} - lim_{x \to 2} 24}{x^{2} - 4}$

Step 3

Move the exponent $3$ from $x^{3}$ outside the limit using the Limits Power Rule.

$\frac{{(lim_{x \to 2} x)}^{4} + {(lim_{x \to 2} x)}^{3} - lim_{x \to 2} 24}{x^{2} - 4}$

Step 4

Evaluate the limit of $24$ which is constant as $x$ approaches $2$ .

$\frac{{(lim_{x \to 2} x)}^{4} + {(lim_{x \to 2} x)}^{3} - 1 \cdot 24}{x^{2} - 4}$

Step 5

Evaluate the limits by plugging in $2$ for all occurrences of $x$ .

Tap for more steps...

Step 5.1

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

$\frac{2^{4} + {(lim_{x \to 2} x)}^{3} - 1 \cdot 24}{x^{2} - 4}$

Step 5.2

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

$\frac{2^{4} + 2^{3} - 1 \cdot 24}{x^{2} - 4}$

Step 6

Simplify the answer.

Tap for more steps...

Step 6.1

Simplify the numerator.

Tap for more steps...

Step 6.1.1

Raise $2$ to the power of $4$ .

$\frac{16 + 2^{3} - 1 \cdot 24}{x^{2} - 4}$

Step 6.1.2

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

$\frac{16 + 8 - 1 \cdot 24}{x^{2} - 4}$

Step 6.1.3

Multiply $- 1$ by $24$ .

$\frac{16 + 8 - 24}{x^{2} - 4}$

Step 6.1.4

Add $16$ and $8$ .

$\frac{24 - 24}{x^{2} - 4}$

Step 6.1.5

Subtract $24$ from $24$ .

$\frac{0}{x^{2} - 4}$

Step 6.2

Simplify the denominator.

Tap for more steps...

Step 6.2.1

Rewrite $4$ as $2^{2}$ .

$\frac{0}{x^{2} - 2^{2}}$

Step 6.2.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 2$ .

$\frac{0}{(x + 2) (x - 2)}$

Step 6.3

Divide $0$ by $(x + 2) (x - 2)$ .

$0$