Calculus Examples

$lim_{x \to \infty} \frac{2 x - 3}{4 x + 5}$

Step 1

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

$lim_{x \to \infty} \frac{\frac{2 x}{x} + \frac{- 3}{x}}{\frac{4 x}{x} + \frac{5}{x}}$

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.

$lim_{x \to \infty} \frac{\frac{2 x}{x} + \frac{- 3}{x}}{\frac{4 x}{x} + \frac{5}{x}}$

Step 2.1.1.2

Divide $2$ by $1$ .

$lim_{x \to \infty} \frac{2 + \frac{- 3}{x}}{\frac{4 x}{x} + \frac{5}{x}}$

Step 2.1.2

Move the negative in front of the fraction.

$lim_{x \to \infty} \frac{2 - \frac{3}{x}}{\frac{4 x}{x} + \frac{5}{x}}$

Step 2.2

Cancel the common factor of $x$ .

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Step 2.2.1

Cancel the common factor.

$lim_{x \to \infty} \frac{2 - \frac{3}{x}}{\frac{4 x}{x} + \frac{5}{x}}$

Step 2.2.2

Divide $4$ by $1$ .

$lim_{x \to \infty} \frac{2 - \frac{3}{x}}{4 + \frac{5}{x}}$

Step 2.3

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

$\frac{lim_{x \to \infty} 2 - \frac{3}{x}}{lim_{x \to \infty} 4 + \frac{5}{x}}$

Step 2.4

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

$\frac{lim_{x \to \infty} 2 - lim_{x \to \infty} \frac{3}{x}}{lim_{x \to \infty} 4 + \frac{5}{x}}$

Step 2.5

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

$\frac{2 - lim_{x \to \infty} \frac{3}{x}}{lim_{x \to \infty} 4 + \frac{5}{x}}$

Step 2.6

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$\frac{2 - 3 lim_{x \to \infty} \frac{1}{x}}{lim_{x \to \infty} 4 + \frac{5}{x}}$

Step 3

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

$\frac{2 - 3 \cdot 0}{lim_{x \to \infty} 4 + \frac{5}{x}}$

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 $\infty$ .

$\frac{2 - 3 \cdot 0}{lim_{x \to \infty} 4 + lim_{x \to \infty} \frac{5}{x}}$

Step 4.2

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

$\frac{2 - 3 \cdot 0}{4 + lim_{x \to \infty} \frac{5}{x}}$

Step 4.3

Move the term $5$ outside of the limit because it is constant with respect to $x$ .

$\frac{2 - 3 \cdot 0}{4 + 5 lim_{x \to \infty} \frac{1}{x}}$

Step 5

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

$\frac{2 - 3 \cdot 0}{4 + 5 \cdot 0}$

Step 6

Simplify the answer.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Multiply $- 3$ by $0$ .

$\frac{2 + 0}{4 + 5 \cdot 0}$

Step 6.1.2

Add $2$ and $0$ .

$\frac{2}{4 + 5 \cdot 0}$

Step 6.2

Simplify the denominator.

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Step 6.2.1

Multiply $5$ by $0$ .

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

Step 6.2.2

Add $4$ and $0$ .

$\frac{2}{4}$

Step 6.3

Cancel the common factor of $2$ and $4$ .

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Step 6.3.1

Factor $2$ out of $2$ .

$\frac{2 (1)}{4}$

Step 6.3.2

Cancel the common factors.

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Step 6.3.2.1

Factor $2$ out of $4$ .

$\frac{2 \cdot 1}{2 \cdot 2}$

Step 6.3.2.2

Cancel the common factor.

$\frac{2 \cdot 1}{2 \cdot 2}$

Step 6.3.2.3

Rewrite the expression.

$\frac{1}{2}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{2}$

Decimal Form:

$0.5$