Calculus Examples

$lim_{x \to \infty} \frac{4 x + 1}{12 x^{2} - 7}$

Step 1

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

$lim_{x \to \infty} \frac{\frac{4 x}{x^{2}} + \frac{1}{x^{2}}}{\frac{12 x^{2}}{x^{2}} + \frac{- 7}{x^{2}}}$

Step 2

Evaluate the limit.

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

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Factor $x$ out of $4 x$ .

$lim_{x \to \infty} \frac{\frac{x \cdot 4}{x^{2}} + \frac{1}{x^{2}}}{\frac{12 x^{2}}{x^{2}} + \frac{- 7}{x^{2}}}$

Cancel the common factors.

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Factor $x$ out of $x^{2}$ .

$lim_{x \to \infty} \frac{\frac{x \cdot 4}{x \cdot x} + \frac{1}{x^{2}}}{\frac{12 x^{2}}{x^{2}} + \frac{- 7}{x^{2}}}$

Cancel the common factor.

$lim_{x \to \infty} \frac{\frac{x \cdot 4}{x \cdot x} + \frac{1}{x^{2}}}{\frac{12 x^{2}}{x^{2}} + \frac{- 7}{x^{2}}}$

Rewrite the expression.

$lim_{x \to \infty} \frac{\frac{4}{x} + \frac{1}{x^{2}}}{\frac{12 x^{2}}{x^{2}} + \frac{- 7}{x^{2}}}$

Simplify each term.

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

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Cancel the common factor.

$lim_{x \to \infty} \frac{\frac{4}{x} + \frac{1}{x^{2}}}{\frac{12 x^{2}}{x^{2}} + \frac{- 7}{x^{2}}}$

Divide $12$ by $1$ .

$lim_{x \to \infty} \frac{\frac{4}{x} + \frac{1}{x^{2}}}{12 + \frac{- 7}{x^{2}}}$

Move the negative in front of the fraction.

$lim_{x \to \infty} \frac{\frac{4}{x} + \frac{1}{x^{2}}}{12 - \frac{7}{x^{2}}}$

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

$\frac{lim_{x \to \infty} \frac{4}{x} + \frac{1}{x^{2}}}{lim_{x \to \infty} 12 - \frac{7}{x^{2}}}$

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

$\frac{lim_{x \to \infty} \frac{4}{x} + lim_{x \to \infty} \frac{1}{x^{2}}}{lim_{x \to \infty} 12 - \frac{7}{x^{2}}}$

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

$\frac{4 lim_{x \to \infty} \frac{1}{x} + lim_{x \to \infty} \frac{1}{x^{2}}}{lim_{x \to \infty} 12 - \frac{7}{x^{2}}}$

Step 3

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

$\frac{4 \cdot 0 + lim_{x \to \infty} \frac{1}{x^{2}}}{lim_{x \to \infty} 12 - \frac{7}{x^{2}}}$

Step 4

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

$\frac{4 \cdot 0 + 0}{lim_{x \to \infty} 12 - \frac{7}{x^{2}}}$

Step 5

Evaluate the limit.

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

$\frac{4 \cdot 0 + 0}{lim_{x \to \infty} 12 - lim_{x \to \infty} \frac{7}{x^{2}}}$

Evaluate the limit of $12$ which is constant as $x$ approaches $\infty$ .

$\frac{4 \cdot 0 + 0}{12 - lim_{x \to \infty} \frac{7}{x^{2}}}$

Move the term $7$ outside of the limit because it is constant with respect to $x$ .

$\frac{4 \cdot 0 + 0}{12 - 7 lim_{x \to \infty} \frac{1}{x^{2}}}$

Step 6

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

$\frac{4 \cdot 0 + 0}{12 - 7 \cdot 0}$

Step 7

Simplify the answer.

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Cancel the common factor of $4 \cdot 0 + 0$ and $12 - 7 \cdot 0$ .

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Rewrite $12$ as $- 1 (- 12)$ .

$\frac{4 \cdot 0 + 0}{- 1 (- 12) - 7 \cdot 0}$

Factor $- 1$ out of $- 7 \cdot 0$ .

$\frac{4 \cdot 0 + 0}{- 1 (- 12) - (7 \cdot 0)}$

Factor $- 1$ out of $- 1 (- 12) - (7 \cdot 0)$ .

$\frac{4 \cdot 0 + 0}{- 1 (- 12 + 7 \cdot 0)}$

Reorder terms.

$\frac{4 \cdot 0 + 0}{- 1 (- 12 + 0 \cdot 7)}$

Factor $12$ out of $4 \cdot 0$ .

$\frac{12 (4 \cdot 0) + 0}{- 1 (- 12 + 0 \cdot 7)}$

Factor $12$ out of $0$ .

$\frac{12 (4 \cdot 0) + 12 (0)}{- 1 (- 12 + 0 \cdot 7)}$

Factor $12$ out of $12 (4 \cdot 0) + 12 (0)$ .

$\frac{12 (4 \cdot 0 + 0)}{- 1 (- 12 + 0 \cdot 7)}$

Cancel the common factors.

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Factor $12$ out of $- 1 (- 12 + 0 \cdot 7)$ .

$\frac{12 (4 \cdot 0 + 0)}{12 (- 1 (- 1 + 0 \cdot 7))}$

Cancel the common factor.

$\frac{12 (4 \cdot 0 + 0)}{12 (- 1 (- 1 + 0 \cdot 7))}$

Rewrite the expression.

$\frac{4 \cdot 0 + 0}{- 1 (- 1 + 0 \cdot 7)}$

Simplify the numerator.

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Multiply $4$ by $0$ .

$\frac{0 + 0}{- 1 (- 1 + 0 \cdot 7)}$

Add $0$ and $0$ .

$\frac{0}{- 1 (- 1 + 0 \cdot 7)}$

Simplify the denominator.

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Multiply $0$ by $7$ .

$\frac{0}{- 1 (- 1 + 0)}$

Add $- 1$ and $0$ .

$\frac{0}{- 1 \cdot - 1}$

Multiply $- 1$ by $- 1$ .

$\frac{0}{1}$

Divide $0$ by $1$ .

$0$