Calculus Examples

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

Step 1

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

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

Step 2

Reduce the expression by cancelling the common factors.

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Step 2.1

Cancel the common factor of $x^{2}$ and $x$ .

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Step 2.1.1

Factor $x$ out of $x^{2}$ .

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

Step 2.1.2

Cancel the common factors.

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Step 2.1.2.1

Raise $x$ to the power of $1$ .

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

Step 2.1.2.2

Factor $x$ out of $x^{1}$ .

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

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

As $x$ 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.

$- \infty$