Calculus Examples

$lim_{x \to \infty} 7 + \frac{1}{3 x} - \frac{2}{x^{2}}$

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 $\infty$ .

$lim_{x \to \infty} 7 + lim_{x \to \infty} \frac{1}{3 x} - lim_{x \to \infty} \frac{2}{x^{2}}$

Step 1.2

Evaluate the limit of $7$ which is constant as $x$ approaches $\infty$ .

$7 + lim_{x \to \infty} \frac{1}{3 x} - lim_{x \to \infty} \frac{2}{x^{2}}$

Step 1.3

Move the term $\frac{1}{3}$ outside of the limit because it is constant with respect to $x$ .

$7 + \frac{1}{3} lim_{x \to \infty} \frac{1}{x} - lim_{x \to \infty} \frac{2}{x^{2}}$

Step 2

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

$7 + \frac{1}{3} \cdot 0 - lim_{x \to \infty} \frac{2}{x^{2}}$

Step 3

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$7 + \frac{1}{3} \cdot 0 - 2 lim_{x \to \infty} \frac{1}{x^{2}}$

Step 4

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

$7 + \frac{1}{3} \cdot 0 - 2 \cdot 0$

Step 5

Simplify the answer.

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Step 5.1

Simplify each term.

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Step 5.1.1

Multiply $\frac{1}{3}$ by $0$ .

$7 + 0 - 2 \cdot 0$

Step 5.1.2

Multiply $- 2$ by $0$ .

$7 + 0 + 0$

Step 5.2

Add $7$ and $0$ .

$7 + 0$

Step 5.3

Add $7$ and $0$ .

$7$