Calculus Examples

lim x \to \infty \frac{1}{2} x - \frac{4}{x^{2}}

$lim_{x \to \infty} \frac{1}{2} x - \frac{4}{x^{2}}$

Step 1

Evaluate the limit.

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Step 1.1

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

x

$x$ approaches

\infty

$\infty$ .

lim x \to \infty \frac{1}{2} x - lim x \to \infty \frac{4}{x^{2}}

$lim_{x \to \infty} \frac{1}{2} x - lim_{x \to \infty} \frac{4}{x^{2}}$

Step 1.2

Combine

\frac{1}{2}

$\frac{1}{2}$ and

x

$x$ .

lim x \to \infty \frac{x}{2} - lim x \to \infty \frac{4}{x^{2}}

$lim_{x \to \infty} \frac{x}{2} - lim_{x \to \infty} \frac{4}{x^{2}}$

Step 1.3

The limit at infinity of a polynomial whose leading coefficient is positive is infinity.

\infty - lim x \to \infty \frac{4}{x^{2}}

$\infty - lim_{x \to \infty} \frac{4}{x^{2}}$

Step 1.4

Move the term

4

$4$ outside of the limit because it is constant with respect to

x

$x$ .

\infty - 4 lim x \to \infty \frac{1}{x^{2}}

$\infty - 4 lim_{x \to \infty} \frac{1}{x^{2}}$

\infty - 4 lim x \to \infty \frac{1}{x^{2}}

$\infty - 4 lim_{x \to \infty} \frac{1}{x^{2}}$

Step 2

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

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

$\frac{1}{x^{2}}$ approaches

0

$0$ .

\infty - 4 \cdot 0

$\infty - 4 \cdot 0$

Step 3

Simplify the answer.

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Step 3.1

Multiply

- 4

$- 4$ by

0

$0$ .

\infty + 0

$\infty + 0$

Step 3.2

Add

\infty

$\infty$ and

0

$0$ .

$\infty$