Calculus Examples

$lim_{x \to \infty} \frac{4 - x^{2}}{4 x^{2} - x - 2}$

Step 1

Divide the numerator and denominator by the highest power of $x$ in the denominator, which is $x^{2}$ .

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

Step 2

Evaluate the limit.

Tap for more steps...

Step 2.1

Simplify each term.

Tap for more steps...

Step 2.1.1

Cancel the common factor of $x^{2}$ .

Tap for more steps...

Step 2.1.1.1

Cancel the common factor.

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

Step 2.1.1.2

Rewrite the expression.

$lim_{x \to \infty} \frac{\frac{4}{x^{2}} - 1 \cdot 1}{\frac{4 x^{2}}{x^{2}} - \frac{x}{x^{2}} + \frac{- 2}{x^{2}}}$

Step 2.1.2

Multiply $- 1$ by $1$ .

$lim_{x \to \infty} \frac{\frac{4}{x^{2}} - 1}{\frac{4 x^{2}}{x^{2}} - \frac{x}{x^{2}} + \frac{- 2}{x^{2}}}$

Step 2.2

Simplify each term.

Tap for more steps...

Step 2.2.1

Cancel the common factor of $x^{2}$ .

Tap for more steps...

Step 2.2.1.1

Cancel the common factor.

$lim_{x \to \infty} \frac{\frac{4}{x^{2}} - 1}{\frac{4 x^{2}}{x^{2}} - \frac{x}{x^{2}} + \frac{- 2}{x^{2}}}$

Step 2.2.1.2

Divide $4$ by $1$ .

$lim_{x \to \infty} \frac{\frac{4}{x^{2}} - 1}{4 - \frac{x}{x^{2}} + \frac{- 2}{x^{2}}}$

Step 2.2.2

Cancel the common factor of $x$ and $x^{2}$ .

Tap for more steps...

Step 2.2.2.1

Raise $x$ to the power of $1$ .

$lim_{x \to \infty} \frac{\frac{4}{x^{2}} - 1}{4 - \frac{x^{1}}{x^{2}} + \frac{- 2}{x^{2}}}$

Step 2.2.2.2

Factor $x$ out of $x^{1}$ .

$lim_{x \to \infty} \frac{\frac{4}{x^{2}} - 1}{4 - \frac{x \cdot 1}{x^{2}} + \frac{- 2}{x^{2}}}$

Step 2.2.2.3

Cancel the common factors.

Tap for more steps...

Step 2.2.2.3.1

Factor $x$ out of $x^{2}$ .

$lim_{x \to \infty} \frac{\frac{4}{x^{2}} - 1}{4 - \frac{x \cdot 1}{x \cdot x} + \frac{- 2}{x^{2}}}$

Step 2.2.2.3.2

Cancel the common factor.

$lim_{x \to \infty} \frac{\frac{4}{x^{2}} - 1}{4 - \frac{x \cdot 1}{x \cdot x} + \frac{- 2}{x^{2}}}$

Step 2.2.2.3.3

Rewrite the expression.

$lim_{x \to \infty} \frac{\frac{4}{x^{2}} - 1}{4 - \frac{1}{x} + \frac{- 2}{x^{2}}}$

Step 2.2.3

Move the negative in front of the fraction.

$lim_{x \to \infty} \frac{\frac{4}{x^{2}} - 1}{4 - \frac{1}{x} - \frac{2}{x^{2}}}$

Step 2.3

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $\infty$ .

$\frac{lim_{x \to \infty} \frac{4}{x^{2}} - 1}{lim_{x \to \infty} 4 - \frac{1}{x} - \frac{2}{x^{2}}}$

Step 2.4

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

$\frac{lim_{x \to \infty} \frac{4}{x^{2}} - lim_{x \to \infty} 1}{lim_{x \to \infty} 4 - \frac{1}{x} - \frac{2}{x^{2}}}$

Step 2.5

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

$\frac{4 lim_{x \to \infty} \frac{1}{x^{2}} - lim_{x \to \infty} 1}{lim_{x \to \infty} 4 - \frac{1}{x} - \frac{2}{x^{2}}}$

Step 3

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{x^{2}}$ approaches $0$ .

$\frac{4 \cdot 0 - lim_{x \to \infty} 1}{lim_{x \to \infty} 4 - \frac{1}{x} - \frac{2}{x^{2}}}$

Step 4

Evaluate the limit.

Tap for more steps...

Step 4.1

Evaluate the limit of $1$ which is constant as $x$ approaches $\infty$ .

$\frac{4 \cdot 0 - 1 \cdot 1}{lim_{x \to \infty} 4 - \frac{1}{x} - \frac{2}{x^{2}}}$

Step 4.2

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

$\frac{4 \cdot 0 - 1 \cdot 1}{lim_{x \to \infty} 4 - lim_{x \to \infty} \frac{1}{x} - lim_{x \to \infty} \frac{2}{x^{2}}}$

Step 4.3

Evaluate the limit of $4$ which is constant as $x$ approaches $\infty$ .

$\frac{4 \cdot 0 - 1 \cdot 1}{4 - lim_{x \to \infty} \frac{1}{x} - lim_{x \to \infty} \frac{2}{x^{2}}}$

Step 5

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{x}$ approaches $0$ .

$\frac{4 \cdot 0 - 1 \cdot 1}{4 - 0 - lim_{x \to \infty} \frac{2}{x^{2}}}$

Step 6

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$\frac{4 \cdot 0 - 1 \cdot 1}{4 - 0 - 2 lim_{x \to \infty} \frac{1}{x^{2}}}$

Step 7

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{x^{2}}$ approaches $0$ .

$\frac{4 \cdot 0 - 1 \cdot 1}{4 - 0 - 2 \cdot 0}$

Step 8

Simplify the answer.

Tap for more steps...

Step 8.1

Simplify the numerator.

Tap for more steps...

Step 8.1.1

Multiply $4$ by $0$ .

$\frac{0 - 1 \cdot 1}{4 - 0 - 2 \cdot 0}$

Step 8.1.2

Multiply $- 1$ by $1$ .

$\frac{0 - 1}{4 - 0 - 2 \cdot 0}$

Step 8.1.3

Subtract $1$ from $0$ .

$\frac{- 1}{4 - 0 - 2 \cdot 0}$

Step 8.2

Simplify the denominator.

Tap for more steps...

Step 8.2.1

Multiply $- 1$ by $0$ .

$\frac{- 1}{4 + 0 - 2 \cdot 0}$

Step 8.2.2

Multiply $- 2$ by $0$ .

$\frac{- 1}{4 + 0 + 0}$

Step 8.2.3

Add $4 + 0$ and $0$ .

$\frac{- 1}{4 + 0}$

Step 8.2.4

Add $4$ and $0$ .

$\frac{- 1}{4}$

Step 8.3

Move the negative in front of the fraction.

$- \frac{1}{4}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$- \frac{1}{4}$

Decimal Form:

$- 0.25$