Calculus Examples

$lim_{x \to \infty} \frac{y x + 6}{2 x - 7} = 4$

Step 1

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

$lim_{x \to \infty} \frac{\frac{y x}{x} + \frac{6}{x}}{\frac{2 x}{x} + \frac{- 7}{x}} = 4$

Step 2

Evaluate the limit.

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Step 2.1

Cancel the common factor of $x$ .

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Step 2.1.1

Cancel the common factor.

$lim_{x \to \infty} \frac{\frac{y x}{x} + \frac{6}{x}}{\frac{2 x}{x} + \frac{- 7}{x}} = 4$

Step 2.1.2

Divide $y$ by $1$ .

$lim_{x \to \infty} \frac{y + \frac{6}{x}}{\frac{2 x}{x} + \frac{- 7}{x}} = 4$

Step 2.2

Simplify each term.

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Step 2.2.1

Cancel the common factor of $x$ .

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Step 2.2.1.1

Cancel the common factor.

$lim_{x \to \infty} \frac{y + \frac{6}{x}}{\frac{2 x}{x} + \frac{- 7}{x}} = 4$

Step 2.2.1.2

Divide $2$ by $1$ .

$lim_{x \to \infty} \frac{y + \frac{6}{x}}{2 + \frac{- 7}{x}} = 4$

Step 2.2.2

Move the negative in front of the fraction.

$lim_{x \to \infty} \frac{y + \frac{6}{x}}{2 - \frac{7}{x}} = 4$

Step 2.3

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

$\frac{lim_{x \to \infty} y + \frac{6}{x}}{lim_{x \to \infty} 2 - \frac{7}{x}} = 4$

Step 2.4

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

$\frac{lim_{x \to \infty} y + lim_{x \to \infty} \frac{6}{x}}{lim_{x \to \infty} 2 - \frac{7}{x}} = 4$

Step 2.5

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

$\frac{y + lim_{x \to \infty} \frac{6}{x}}{lim_{x \to \infty} 2 - \frac{7}{x}} = 4$

Step 2.6

Move the term $6$ outside of the limit because it is constant with respect to $x$ .

$\frac{y + 6 lim_{x \to \infty} \frac{1}{x}}{lim_{x \to \infty} 2 - \frac{7}{x}} = 4$

Step 3

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

$\frac{y + 6 \cdot 0}{lim_{x \to \infty} 2 - \frac{7}{x}} = 4$

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 $\infty$ .

$\frac{y + 6 \cdot 0}{lim_{x \to \infty} 2 - lim_{x \to \infty} \frac{7}{x}} = 4$

Step 4.2

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

$\frac{y + 6 \cdot 0}{2 - lim_{x \to \infty} \frac{7}{x}} = 4$

Step 4.3

Move the term $7$ outside of the limit because it is constant with respect to $x$ .

$\frac{y + 6 \cdot 0}{2 - 7 lim_{x \to \infty} \frac{1}{x}} = 4$

Step 5

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

$\frac{y + 6 \cdot 0}{2 - 7 \cdot 0} = 4$

Step 6

Simplify the answer.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Multiply $6$ by $0$ .

$\frac{y + 0}{2 - 7 \cdot 0} = 4$

Step 6.1.2

Add $y$ and $0$ .

$\frac{y}{2 - 7 \cdot 0} = 4$

Step 6.2

Simplify the denominator.

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Step 6.2.1

Multiply $- 7$ by $0$ .

$\frac{y}{2 + 0} = 4$

Step 6.2.2

Add $2$ and $0$ .

$\frac{y}{2} = 4$