Calculus Examples

$lim_{x \to \infty} \frac{3 x - 1}{2 - x}$

Step 1

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

$lim_{x \to \infty} \frac{\frac{3 x}{x} - \frac{1}{x}}{\frac{2}{x} - \frac{x}{x}}$

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.

$lim_{x \to \infty} \frac{\frac{3 x}{x} - \frac{1}{x}}{\frac{2}{x} - \frac{x}{x}}$

Step 2.1.2

Divide $3$ by $1$ .

$lim_{x \to \infty} \frac{3 - \frac{1}{x}}{\frac{2}{x} - \frac{x}{x}}$

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.

$lim_{x \to \infty} \frac{3 - \frac{1}{x}}{\frac{2}{x} - \frac{x}{x}}$

Step 2.2.1.2

Rewrite the expression.

$lim_{x \to \infty} \frac{3 - \frac{1}{x}}{\frac{2}{x} - 1 \cdot 1}$

Step 2.2.2

Multiply $- 1$ by $1$ .

$lim_{x \to \infty} \frac{3 - \frac{1}{x}}{\frac{2}{x} - 1}$

Step 2.3

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

$\frac{lim_{x \to \infty} 3 - \frac{1}{x}}{lim_{x \to \infty} \frac{2}{x} - 1}$

Step 2.4

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

$\frac{lim_{x \to \infty} 3 - lim_{x \to \infty} \frac{1}{x}}{lim_{x \to \infty} \frac{2}{x} - 1}$

Step 2.5

Evaluate the limit of $3$ which is constant as $x$ approaches $\infty$ .

$\frac{3 - lim_{x \to \infty} \frac{1}{x}}{lim_{x \to \infty} \frac{2}{x} - 1}$

Step 3

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

$\frac{3 - 0}{lim_{x \to \infty} \frac{2}{x} - 1}$

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 $\infty$ .

$\frac{3 - 0}{lim_{x \to \infty} \frac{2}{x} - lim_{x \to \infty} 1}$

Step 4.2

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

$\frac{3 - 0}{2 lim_{x \to \infty} \frac{1}{x} - lim_{x \to \infty} 1}$

Step 5

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

$\frac{3 - 0}{2 \cdot 0 - lim_{x \to \infty} 1}$

Step 6

Evaluate the limit.

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Step 6.1

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

$\frac{3 - 0}{2 \cdot 0 - 1 \cdot 1}$

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 $0$ .

$\frac{3 + 0}{2 \cdot 0 - 1 \cdot 1}$

Step 6.2.1.2

Add $3$ and $0$ .

$\frac{3}{2 \cdot 0 - 1 \cdot 1}$

Step 6.2.2

Simplify the denominator.

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Step 6.2.2.1

Multiply $2$ by $0$ .

$\frac{3}{0 - 1 \cdot 1}$

Step 6.2.2.2

Multiply $- 1$ by $1$ .

$\frac{3}{0 - 1}$

Step 6.2.2.3

Subtract $1$ from $0$ .

$\frac{3}{- 1}$

Step 6.2.3

Divide $3$ by $- 1$ .

$- 3$