Calculus Examples

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

Step 1

Divide the numerator and denominator by the highest power of $x$ in the denominator, which is $x^{2}$ .

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

Step 2

Evaluate the limit.

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Step 2.1

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

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Step 2.1.1

Cancel the common factor.

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

Step 2.1.2

Rewrite the expression.

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

Step 2.2

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

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Step 2.2.1

Cancel the common factor.

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

Step 2.2.2

Rewrite the expression.

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

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

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

Step 3

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

$\frac{1}{1 + 0}$

Step 4

Simplify the answer.

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Step 4.1

Add $1$ and $0$ .

$\frac{1}{1}$

Step 4.2

Divide $1$ by $1$ .

$1$