Calculus Examples

lim x \to \infty \frac{1 - x^{2}}{2 x}

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

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{1}{x} - \frac{x^{2}}{x}}{\frac{2 x}{x}}

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

Step 2

Reduce the expression by cancelling the common factors.

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Step 2.1

Cancel the common factor of

x^{2}

$x^{2}$ and

x

$x$ .

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Step 2.1.1

Factor

x

$x$ out of

x^{2}

$x^{2}$ .

lim x \to \infty \frac{\frac{1}{x} - \frac{x \cdot x}{x}}{\frac{2 x}{x}}

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

Step 2.1.2

Cancel the common factors.

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Step 2.1.2.1

Raise

x

$x$ to the power of

1

$1$ .

lim x \to \infty \frac{\frac{1}{x} - \frac{x \cdot x}{x^{1}}}{\frac{2 x}{x}}

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

Step 2.1.2.2

Factor

x

$x$ out of

x^{1}

$x^{1}$ .

lim x \to \infty \frac{\frac{1}{x} - \frac{x \cdot x}{x \cdot 1}}{\frac{2 x}{x}}

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

Step 2.1.2.3

Cancel the common factor.

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

Step 2.1.2.4

Rewrite the expression.

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

Step 2.1.2.5

Divide

$x$ by

$1$ .

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

Step 2.2

Cancel the common factor of

$x$ .

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Step 2.2.1

Cancel the common factor.

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

Step 2.2.2

Divide

$2$ by

$1$ .

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

Step 3

$x$ approaches

$\infty$ , the fraction

$\frac{1}{x}$ approaches

$0$ .

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

Step 4

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

$\frac{0 - x}{2}$ approaches negative infinity.

$- \infty$