Calculus Examples

$lim_{x \to \infty} \frac{1 - x}{2 + 5 x}$

Step 1

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

$lim_{x \to \infty} \frac{\frac{1}{x} - \frac{x}{x}}{\frac{2}{x} + \frac{5 x}{x}}$

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 $x$ .

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Step 2.1.1.1

Cancel the common factor.

$lim_{x \to \infty} \frac{\frac{1}{x} - \frac{x}{x}}{\frac{2}{x} + \frac{5 x}{x}}$

Step 2.1.1.2

Rewrite the expression.

$lim_{x \to \infty} \frac{\frac{1}{x} - 1 \cdot 1}{\frac{2}{x} + \frac{5 x}{x}}$

Step 2.1.2

Multiply $- 1$ by $1$ .

$lim_{x \to \infty} \frac{\frac{1}{x} - 1}{\frac{2}{x} + \frac{5 x}{x}}$

Step 2.2

Cancel the common factor of $x$ .

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Step 2.2.1

Cancel the common factor.

$lim_{x \to \infty} \frac{\frac{1}{x} - 1}{\frac{2}{x} + \frac{5 x}{x}}$

Step 2.2.2

Divide $5$ by $1$ .

$lim_{x \to \infty} \frac{\frac{1}{x} - 1}{\frac{2}{x} + 5}$

Step 2.3

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

$\frac{lim_{x \to \infty} \frac{1}{x} - 1}{lim_{x \to \infty} \frac{2}{x} + 5}$

Step 2.4

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

$\frac{lim_{x \to \infty} \frac{1}{x} - lim_{x \to \infty} 1}{lim_{x \to \infty} \frac{2}{x} + 5}$

Step 3

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

$\frac{0 - lim_{x \to \infty} 1}{lim_{x \to \infty} \frac{2}{x} + 5}$

Step 4

Evaluate the limit.

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Step 4.1

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

$\frac{0 - 1 \cdot 1}{lim_{x \to \infty} \frac{2}{x} + 5}$

Step 4.2

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

$\frac{0 - 1 \cdot 1}{lim_{x \to \infty} \frac{2}{x} + lim_{x \to \infty} 5}$

Step 4.3

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

$\frac{0 - 1 \cdot 1}{2 lim_{x \to \infty} \frac{1}{x} + lim_{x \to \infty} 5}$

Step 5

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

$\frac{0 - 1 \cdot 1}{2 \cdot 0 + lim_{x \to \infty} 5}$

Step 6

Evaluate the limit.

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Step 6.1

Evaluate the limit of $5$ which is constant as $x$ approaches $\infty$ .

$\frac{0 - 1 \cdot 1}{2 \cdot 0 + 5}$

Step 6.2

Simplify the answer.

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Step 6.2.1

Simplify the numerator.

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Step 6.2.1.1

Multiply $- 1$ by $1$ .

$\frac{0 - 1}{2 \cdot 0 + 5}$

Step 6.2.1.2

Subtract $1$ from $0$ .

$\frac{- 1}{2 \cdot 0 + 5}$

Step 6.2.2

Simplify the denominator.

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Step 6.2.2.1

Multiply $2$ by $0$ .

$\frac{- 1}{0 + 5}$

Step 6.2.2.2

Add $0$ and $5$ .

$\frac{- 1}{5}$

Step 6.2.3

Move the negative in front of the fraction.

$- \frac{1}{5}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$- \frac{1}{5}$

Decimal Form:

$- 0.2$