Calculus Examples

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

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 $x$ approaches $\infty$ .

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

Step 1.2

Combine $\frac{1}{2}$ and $x$ .

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

Step 1.3

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

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

Step 1.4

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

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

Step 2

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{x^{2}}$ approaches $0$ .

$\infty - 4 \cdot 0$

Step 3

Simplify the answer.

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Step 3.1

Multiply $- 4$ by $0$ .

$\infty + 0$

Step 3.2

Add $\infty$ and $0$ .

$\infty$