Calculus Examples

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

Step 1

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

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

Step 2

Evaluate the limit.

Tap for more steps...

Step 2.1

Cancel the common factor of $x$ .

Tap for more steps...

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.

Tap for more steps...

Step 2.2.1

Cancel the common factor of $x$ .

Tap for more steps...

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.

Tap for more steps...

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:

$0.5$