Calculus Examples

lim x \to - \infty \frac{x}{2 x - 3}

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

Step 1

Divide the numerator and denominator by the highest power of

x

$x$ in the denominator, which is

x

$x$ .

lim x \to - \infty \frac{\frac{x}{x}}{\frac{2 x}{x} + \frac{- 3}{x}}

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

Step 2

Evaluate the limit.

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Step 2.1

Cancel the common factor of

x

$x$ .

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Step 2.1.1

Cancel the common factor.

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

Step 2.1.2

Rewrite the expression.

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

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.

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

Step 2.2.1.2

Divide

$2$ by

$1$ .

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

Step 2.2.2

Move the negative in front of the fraction.

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

Step 2.3

Split the limit using the Limits Quotient Rule on the limit as

$x$ approaches

$- \infty$ .

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

Step 2.4

Evaluate the limit of

$1$ which is constant as

$x$ approaches

$- \infty$ .

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

Step 2.5

Split the limit using the Sum of Limits Rule on the limit as

$x$ approaches

$- \infty$ .

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

Step 2.6

Evaluate the limit of

$2$ which is constant as

$x$ approaches

$- \infty$ .

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

Step 2.7

Move the term

$3$ outside of the limit because it is constant with respect to

$x$ .

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

Step 3

Since its numerator approaches a real number while its denominator is unbounded, the fraction

$\frac{1}{x}$ approaches

$0$ .

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

Step 4

Simplify the denominator.

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Step 4.1

Multiply

$- 3$ by

$0$ .

$\frac{1}{2 + 0}$

Step 4.2

Add

$2$ and

$0$ .

$\frac{1}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{2}$

Decimal Form:

$0.5$