Calculus Examples

$lim_{x \to \infty} \frac{2 x^{2} + 5}{x^{2}}$

Step 1

Evaluate the limit.

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Step 1.1

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

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Step 1.1.1

Cancel the common factor.

$lim_{x \to \infty} \frac{2 + \frac{5}{x^{2}}}{\frac{x^{2}}{x^{2}}}$

Step 1.1.2

Rewrite the expression.

$lim_{x \to \infty} \frac{2 + \frac{5}{x^{2}}}{1}$

Step 1.2

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $\infty$ .

$\frac{lim_{x \to \infty} 2 + \frac{5}{x^{2}}}{lim_{x \to \infty} 1}$

Step 1.3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $\infty$ .

$\frac{lim_{x \to \infty} 2 + lim_{x \to \infty} \frac{5}{x^{2}}}{lim_{x \to \infty} 1}$

Step 1.4

Evaluate the limit of $2$ which is constant as $x$ approaches $\infty$ .

$\frac{2 + lim_{x \to \infty} \frac{5}{x^{2}}}{lim_{x \to \infty} 1}$

Step 1.5

Move the term $5$ outside of the limit because it is constant with respect to $x$ .

$\frac{2 + 5 lim_{x \to \infty} \frac{1}{x^{2}}}{lim_{x \to \infty} 1}$

Step 2

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

$\frac{2 + 5 \cdot 0}{lim_{x \to \infty} 1}$

Step 3

Evaluate the limit.

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Step 3.1

Evaluate the limit of $1$ which is constant as $x$ approaches $\infty$ .

$\frac{2 + 5 \cdot 0}{1}$

Step 3.2

Simplify the answer.

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Step 3.2.1

Divide $2 + 5 \cdot 0$ by $1$ .

$2 + 5 \cdot 0$

Step 3.2.2

Multiply $5$ by $0$ .

$2 + 0$

Step 3.2.3

Add $2$ and $0$ .

$2$