Calculus Examples

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

Step 1

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

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

Step 2

Evaluate the limit.

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Step 2.1

Simplify each term.

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Step 2.1.1

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

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Step 2.1.1.1

Cancel the common factor.

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

Step 2.1.1.2

Rewrite the expression.

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

Step 2.1.2

Multiply $- 1$ by $1$ .

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

Step 2.1.3

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

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Step 2.1.3.1

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

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

Step 2.1.3.2

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

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

Step 2.1.3.3

Cancel the common factors.

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Step 2.1.3.3.1

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

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

Step 2.1.3.3.2

Cancel the common factor.

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

Step 2.1.3.3.3

Rewrite the expression.

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

Step 2.2

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

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Step 2.2.1

Cancel the common factor.

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

Step 2.2.2

Divide $4$ by $1$ .

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

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

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

Step 3

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

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

Step 4

Evaluate the limit.

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Step 4.1

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

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

Step 4.2

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

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

Step 5

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

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

Step 6

Simplify the answer.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Multiply $- 1$ by $0$ .

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

Step 6.1.2

Add $- 1$ and $0$ .

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

Step 6.2

Add $4$ and $0$ .

$\frac{- 1}{4}$

Step 6.3

Move the negative in front of the fraction.

$- \frac{1}{4}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$- \frac{1}{4}$

Decimal Form:

$- 0.25$