Calculus Examples

$lim_{n \to \infty} \frac{6 n^{3} + 2 n^{2} + 10}{n^{3} + 1}$

Step 1

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

$lim_{n \to \infty} \frac{\frac{6 n^{3}}{n^{3}} + \frac{2 n^{2}}{n^{3}} + \frac{10}{n^{3}}}{\frac{n^{3}}{n^{3}} + \frac{1}{n^{3}}}$

Step 2

Evaluate the limit.

Tap for more steps...

Step 2.1

Simplify each term.

Tap for more steps...

Step 2.1.1

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

Tap for more steps...

Step 2.1.1.1

Cancel the common factor.

$lim_{n \to \infty} \frac{\frac{6 n^{3}}{n^{3}} + \frac{2 n^{2}}{n^{3}} + \frac{10}{n^{3}}}{\frac{n^{3}}{n^{3}} + \frac{1}{n^{3}}}$

Step 2.1.1.2

Divide $6$ by $1$ .

$lim_{n \to \infty} \frac{6 + \frac{2 n^{2}}{n^{3}} + \frac{10}{n^{3}}}{\frac{n^{3}}{n^{3}} + \frac{1}{n^{3}}}$

Step 2.1.2

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

Tap for more steps...

Step 2.1.2.1

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

$lim_{n \to \infty} \frac{6 + \frac{n^{2} \cdot 2}{n^{3}} + \frac{10}{n^{3}}}{\frac{n^{3}}{n^{3}} + \frac{1}{n^{3}}}$

Step 2.1.2.2

Cancel the common factors.

Tap for more steps...

Step 2.1.2.2.1

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

$lim_{n \to \infty} \frac{6 + \frac{n^{2} \cdot 2}{n^{2} n} + \frac{10}{n^{3}}}{\frac{n^{3}}{n^{3}} + \frac{1}{n^{3}}}$

Step 2.1.2.2.2

Cancel the common factor.

$lim_{n \to \infty} \frac{6 + \frac{n^{2} \cdot 2}{n^{2} n} + \frac{10}{n^{3}}}{\frac{n^{3}}{n^{3}} + \frac{1}{n^{3}}}$

Step 2.1.2.2.3

Rewrite the expression.

$lim_{n \to \infty} \frac{6 + \frac{2}{n} + \frac{10}{n^{3}}}{\frac{n^{3}}{n^{3}} + \frac{1}{n^{3}}}$

Step 2.2

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

Tap for more steps...

Step 2.2.1

Cancel the common factor.

$lim_{n \to \infty} \frac{6 + \frac{2}{n} + \frac{10}{n^{3}}}{\frac{n^{3}}{n^{3}} + \frac{1}{n^{3}}}$

Step 2.2.2

Rewrite the expression.

$lim_{n \to \infty} \frac{6 + \frac{2}{n} + \frac{10}{n^{3}}}{1 + \frac{1}{n^{3}}}$

Step 2.3

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

$\frac{lim_{n \to \infty} 6 + \frac{2}{n} + \frac{10}{n^{3}}}{lim_{n \to \infty} 1 + \frac{1}{n^{3}}}$

Step 2.4

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

$\frac{lim_{n \to \infty} 6 + lim_{n \to \infty} \frac{2}{n} + lim_{n \to \infty} \frac{10}{n^{3}}}{lim_{n \to \infty} 1 + \frac{1}{n^{3}}}$

Step 2.5

Evaluate the limit of $6$ which is constant as $n$ approaches $\infty$ .

$\frac{6 + lim_{n \to \infty} \frac{2}{n} + lim_{n \to \infty} \frac{10}{n^{3}}}{lim_{n \to \infty} 1 + \frac{1}{n^{3}}}$

Step 2.6

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

$\frac{6 + 2 lim_{n \to \infty} \frac{1}{n} + lim_{n \to \infty} \frac{10}{n^{3}}}{lim_{n \to \infty} 1 + \frac{1}{n^{3}}}$

Step 3

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

$\frac{6 + 2 \cdot 0 + lim_{n \to \infty} \frac{10}{n^{3}}}{lim_{n \to \infty} 1 + \frac{1}{n^{3}}}$

Step 4

Move the term $10$ outside of the limit because it is constant with respect to $n$ .

$\frac{6 + 2 \cdot 0 + 10 lim_{n \to \infty} \frac{1}{n^{3}}}{lim_{n \to \infty} 1 + \frac{1}{n^{3}}}$

Step 5

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

$\frac{6 + 2 \cdot 0 + 10 \cdot 0}{lim_{n \to \infty} 1 + \frac{1}{n^{3}}}$

Step 6

Evaluate the limit.

Tap for more steps...

Step 6.1

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

$\frac{6 + 2 \cdot 0 + 10 \cdot 0}{lim_{n \to \infty} 1 + lim_{n \to \infty} \frac{1}{n^{3}}}$

Step 6.2

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

$\frac{6 + 2 \cdot 0 + 10 \cdot 0}{1 + lim_{n \to \infty} \frac{1}{n^{3}}}$

Step 7

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

$\frac{6 + 2 \cdot 0 + 10 \cdot 0}{1 + 0}$

Step 8

Simplify the answer.

Tap for more steps...

Step 8.1

Simplify the numerator.

Tap for more steps...

Step 8.1.1

Multiply $2$ by $0$ .

$\frac{6 + 0 + 10 \cdot 0}{1 + 0}$

Step 8.1.2

Multiply $10$ by $0$ .

$\frac{6 + 0 + 0}{1 + 0}$

Step 8.1.3

Add $6 + 0$ and $0$ .

$\frac{6 + 0}{1 + 0}$

Step 8.1.4

Add $6$ and $0$ .

$\frac{6}{1 + 0}$

Step 8.2

Add $1$ and $0$ .

$\frac{6}{1}$

Step 8.3

Divide $6$ by $1$ .

$6$