Precalculus Examples

$lim_{n \to \infty} \frac{4 n^{3} + 7 n^{2}}{5 n^{3} - 7 n^{2} + 3}$

Step 1

Divide the numerator and denominator by the highest power of $n$ in the denominator, which is $n^{3}$ .

$lim_{n \to \infty} \frac{\frac{4 n^{3}}{n^{3}} + \frac{7 n^{2}}{n^{3}}}{\frac{5 n^{3}}{n^{3}} + \frac{- 7 n^{2}}{n^{3}} + \frac{3}{n^{3}}}$

Step 2

Evaluate the limit.

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Step 2.1

Simplify each term.

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Step 2.1.1

Cancel the common factor of $n^{3}$ .

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Step 2.1.1.1

Cancel the common factor.

$lim_{n \to \infty} \frac{\frac{4 n^{3}}{n^{3}} + \frac{7 n^{2}}{n^{3}}}{\frac{5 n^{3}}{n^{3}} + \frac{- 7 n^{2}}{n^{3}} + \frac{3}{n^{3}}}$

Step 2.1.1.2

Divide $4$ by $1$ .

$lim_{n \to \infty} \frac{4 + \frac{7 n^{2}}{n^{3}}}{\frac{5 n^{3}}{n^{3}} + \frac{- 7 n^{2}}{n^{3}} + \frac{3}{n^{3}}}$

Step 2.1.2

Cancel the common factor of $n^{2}$ and $n^{3}$ .

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Step 2.1.2.1

Factor $n^{2}$ out of $7 n^{2}$ .

$lim_{n \to \infty} \frac{4 + \frac{n^{2} \cdot 7}{n^{3}}}{\frac{5 n^{3}}{n^{3}} + \frac{- 7 n^{2}}{n^{3}} + \frac{3}{n^{3}}}$

Step 2.1.2.2

Cancel the common factors.

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Step 2.1.2.2.1

Factor $n^{2}$ out of $n^{3}$ .

$lim_{n \to \infty} \frac{4 + \frac{n^{2} \cdot 7}{n^{2} n}}{\frac{5 n^{3}}{n^{3}} + \frac{- 7 n^{2}}{n^{3}} + \frac{3}{n^{3}}}$

Step 2.1.2.2.2

Cancel the common factor.

$lim_{n \to \infty} \frac{4 + \frac{n^{2} \cdot 7}{n^{2} n}}{\frac{5 n^{3}}{n^{3}} + \frac{- 7 n^{2}}{n^{3}} + \frac{3}{n^{3}}}$

Step 2.1.2.2.3

Rewrite the expression.

$lim_{n \to \infty} \frac{4 + \frac{7}{n}}{\frac{5 n^{3}}{n^{3}} + \frac{- 7 n^{2}}{n^{3}} + \frac{3}{n^{3}}}$

Step 2.2

Simplify each term.

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Step 2.2.1

Cancel the common factor of $n^{3}$ .

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Step 2.2.1.1

Cancel the common factor.

$lim_{n \to \infty} \frac{4 + \frac{7}{n}}{\frac{5 n^{3}}{n^{3}} + \frac{- 7 n^{2}}{n^{3}} + \frac{3}{n^{3}}}$

Step 2.2.1.2

Divide $5$ by $1$ .

$lim_{n \to \infty} \frac{4 + \frac{7}{n}}{5 + \frac{- 7 n^{2}}{n^{3}} + \frac{3}{n^{3}}}$

Step 2.2.2

Cancel the common factor of $n^{2}$ and $n^{3}$ .

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Step 2.2.2.1

Factor $n^{2}$ out of $- 7 n^{2}$ .

$lim_{n \to \infty} \frac{4 + \frac{7}{n}}{5 + \frac{n^{2} \cdot - 7}{n^{3}} + \frac{3}{n^{3}}}$

Step 2.2.2.2

Cancel the common factors.

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Step 2.2.2.2.1

Factor $n^{2}$ out of $n^{3}$ .

$lim_{n \to \infty} \frac{4 + \frac{7}{n}}{5 + \frac{n^{2} \cdot - 7}{n^{2} n} + \frac{3}{n^{3}}}$

Step 2.2.2.2.2

Cancel the common factor.

$lim_{n \to \infty} \frac{4 + \frac{7}{n}}{5 + \frac{n^{2} \cdot - 7}{n^{2} n} + \frac{3}{n^{3}}}$

Step 2.2.2.2.3

Rewrite the expression.

$lim_{n \to \infty} \frac{4 + \frac{7}{n}}{5 + \frac{- 7}{n} + \frac{3}{n^{3}}}$

Step 2.2.3

Move the negative in front of the fraction.

$lim_{n \to \infty} \frac{4 + \frac{7}{n}}{5 - \frac{7}{n} + \frac{3}{n^{3}}}$

Step 2.3

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

$\frac{lim_{n \to \infty} 4 + \frac{7}{n}}{lim_{n \to \infty} 5 - \frac{7}{n} + \frac{3}{n^{3}}}$

Step 2.4

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

$\frac{lim_{n \to \infty} 4 + lim_{n \to \infty} \frac{7}{n}}{lim_{n \to \infty} 5 - \frac{7}{n} + \frac{3}{n^{3}}}$

Step 2.5

Evaluate the limit of $4$ which is constant as $n$ approaches $\infty$ .

$\frac{4 + lim_{n \to \infty} \frac{7}{n}}{lim_{n \to \infty} 5 - \frac{7}{n} + \frac{3}{n^{3}}}$

Step 2.6

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

$\frac{4 + 7 lim_{n \to \infty} \frac{1}{n}}{lim_{n \to \infty} 5 - \frac{7}{n} + \frac{3}{n^{3}}}$

Step 3

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

$\frac{4 + 7 \cdot 0}{lim_{n \to \infty} 5 - \frac{7}{n} + \frac{3}{n^{3}}}$

Step 4

Evaluate the limit.

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Step 4.1

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

$\frac{4 + 7 \cdot 0}{lim_{n \to \infty} 5 - lim_{n \to \infty} \frac{7}{n} + lim_{n \to \infty} \frac{3}{n^{3}}}$

Step 4.2

Evaluate the limit of $5$ which is constant as $n$ approaches $\infty$ .

$\frac{4 + 7 \cdot 0}{5 - lim_{n \to \infty} \frac{7}{n} + lim_{n \to \infty} \frac{3}{n^{3}}}$

Step 4.3

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

$\frac{4 + 7 \cdot 0}{5 - 7 lim_{n \to \infty} \frac{1}{n} + lim_{n \to \infty} \frac{3}{n^{3}}}$

Step 5

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

$\frac{4 + 7 \cdot 0}{5 - 7 \cdot 0 + lim_{n \to \infty} \frac{3}{n^{3}}}$

Step 6

Move the term $3$ outside of the limit because it is constant with respect to $n$ .

$\frac{4 + 7 \cdot 0}{5 - 7 \cdot 0 + 3 lim_{n \to \infty} \frac{1}{n^{3}}}$

Step 7

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{n^{3}}$ approaches $0$ .

$\frac{4 + 7 \cdot 0}{5 - 7 \cdot 0 + 3 \cdot 0}$

Step 8

Simplify the answer.

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Step 8.1

Simplify the numerator.

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Step 8.1.1

Multiply $7$ by $0$ .

$\frac{4 + 0}{5 - 7 \cdot 0 + 3 \cdot 0}$

Step 8.1.2

Add $4$ and $0$ .

$\frac{4}{5 - 7 \cdot 0 + 3 \cdot 0}$

Step 8.2

Simplify the denominator.

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Step 8.2.1

Multiply $- 7$ by $0$ .

$\frac{4}{5 + 0 + 3 \cdot 0}$

Step 8.2.2

Multiply $3$ by $0$ .

$\frac{4}{5 + 0 + 0}$

Step 8.2.3

Add $5 + 0$ and $0$ .

$\frac{4}{5 + 0}$

Step 8.2.4

Add $5$ and $0$ .

$\frac{4}{5}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\frac{4}{5}$

Decimal Form:

$0.8$