Calculus Examples

$lim_{n \to 8} \frac{x (n + 1) {(2 n + 1)}^{2}}{{(2 n + 3)}^{2} n}$

Step 1

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

$x lim_{n \to 8} \frac{(n + 1) {(2 n + 1)}^{2}}{{(2 n + 3)}^{2} n}$

Step 2

Split the limit using the Limits Quotient Rule on the limit as $n$ approaches $8$ .

$x \frac{lim_{n \to 8} (n + 1) {(2 n + 1)}^{2}}{lim_{n \to 8} {(2 n + 3)}^{2} n}$

Step 3

Split the limit using the Product of Limits Rule on the limit as $n$ approaches $8$ .

$x \frac{lim_{n \to 8} n + 1 \cdot lim_{n \to 8} {(2 n + 1)}^{2}}{lim_{n \to 8} {(2 n + 3)}^{2} n}$

Step 4

Split the limit using the Sum of Limits Rule on the limit as $n$ approaches $8$ .

$x \frac{(lim_{n \to 8} n + lim_{n \to 8} 1) \cdot lim_{n \to 8} {(2 n + 1)}^{2}}{lim_{n \to 8} {(2 n + 3)}^{2} n}$

Step 5

Evaluate the limit of $1$ which is constant as $n$ approaches $8$ .

$x \frac{(lim_{n \to 8} n + 1) \cdot lim_{n \to 8} {(2 n + 1)}^{2}}{lim_{n \to 8} {(2 n + 3)}^{2} n}$

Step 6

Move the exponent $2$ from ${(2 n + 1)}^{2}$ outside the limit using the Limits Power Rule.

$x \frac{(lim_{n \to 8} n + 1) \cdot {(lim_{n \to 8} 2 n + 1)}^{2}}{lim_{n \to 8} {(2 n + 3)}^{2} n}$

Step 7

Split the limit using the Sum of Limits Rule on the limit as $n$ approaches $8$ .

$x \frac{(lim_{n \to 8} n + 1) \cdot {(lim_{n \to 8} 2 n + lim_{n \to 8} 1)}^{2}}{lim_{n \to 8} {(2 n + 3)}^{2} n}$

Step 8

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

$x \frac{(lim_{n \to 8} n + 1) \cdot {(2 lim_{n \to 8} n + lim_{n \to 8} 1)}^{2}}{lim_{n \to 8} {(2 n + 3)}^{2} n}$

Step 9

Evaluate the limit of $1$ which is constant as $n$ approaches $8$ .

$x \frac{(lim_{n \to 8} n + 1) \cdot {(2 lim_{n \to 8} n + 1)}^{2}}{lim_{n \to 8} {(2 n + 3)}^{2} n}$

Step 10

Split the limit using the Product of Limits Rule on the limit as $n$ approaches $8$ .

$x \frac{(lim_{n \to 8} n + 1) \cdot {(2 lim_{n \to 8} n + 1)}^{2}}{lim_{n \to 8} {(2 n + 3)}^{2} \cdot lim_{n \to 8} n}$

Step 11

Move the exponent $2$ from ${(2 n + 3)}^{2}$ outside the limit using the Limits Power Rule.

$x \frac{(lim_{n \to 8} n + 1) \cdot {(2 lim_{n \to 8} n + 1)}^{2}}{{(lim_{n \to 8} 2 n + 3)}^{2} \cdot lim_{n \to 8} n}$

Step 12

Split the limit using the Sum of Limits Rule on the limit as $n$ approaches $8$ .

$x \frac{(lim_{n \to 8} n + 1) \cdot {(2 lim_{n \to 8} n + 1)}^{2}}{{(lim_{n \to 8} 2 n + lim_{n \to 8} 3)}^{2} \cdot lim_{n \to 8} n}$

Step 13

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

$x \frac{(lim_{n \to 8} n + 1) \cdot {(2 lim_{n \to 8} n + 1)}^{2}}{{(2 lim_{n \to 8} n + lim_{n \to 8} 3)}^{2} \cdot lim_{n \to 8} n}$

Step 14

Evaluate the limit of $3$ which is constant as $n$ approaches $8$ .

$x \frac{(lim_{n \to 8} n + 1) \cdot {(2 lim_{n \to 8} n + 1)}^{2}}{{(2 lim_{n \to 8} n + 3)}^{2} \cdot lim_{n \to 8} n}$

Step 15

Evaluate the limits by plugging in $8$ for all occurrences of $n$ .

Tap for more steps...

Step 15.1

Evaluate the limit of $n$ by plugging in $8$ for $n$ .

$x \frac{(8 + 1) \cdot {(2 lim_{n \to 8} n + 1)}^{2}}{{(2 lim_{n \to 8} n + 3)}^{2} \cdot lim_{n \to 8} n}$

Step 15.2

Evaluate the limit of $n$ by plugging in $8$ for $n$ .

$x \frac{(8 + 1) \cdot {(2 \cdot 8 + 1)}^{2}}{{(2 lim_{n \to 8} n + 3)}^{2} \cdot lim_{n \to 8} n}$

Step 15.3

Evaluate the limit of $n$ by plugging in $8$ for $n$ .

$x \frac{(8 + 1) \cdot {(2 \cdot 8 + 1)}^{2}}{{(2 \cdot 8 + 3)}^{2} \cdot lim_{n \to 8} n}$

Step 15.4

Evaluate the limit of $n$ by plugging in $8$ for $n$ .

$x \frac{(8 + 1) \cdot {(2 \cdot 8 + 1)}^{2}}{{(2 \cdot 8 + 3)}^{2} \cdot 8}$

Step 16

Simplify the answer.

Tap for more steps...

Step 16.1

Simplify the numerator.

Tap for more steps...

Step 16.1.1

Add $8$ and $1$ .

$x \frac{9 {(2 \cdot 8 + 1)}^{2}}{{(2 \cdot 8 + 3)}^{2} \cdot 8}$

Step 16.1.2

Multiply $2$ by $8$ .

$x \frac{9 {(16 + 1)}^{2}}{{(2 \cdot 8 + 3)}^{2} \cdot 8}$

Step 16.1.3

Add $16$ and $1$ .

$x \frac{9 \cdot 17^{2}}{{(2 \cdot 8 + 3)}^{2} \cdot 8}$

Step 16.1.4

Raise $17$ to the power of $2$ .

$x \frac{9 \cdot 289}{{(2 \cdot 8 + 3)}^{2} \cdot 8}$

Step 16.2

Simplify the denominator.

Tap for more steps...

Step 16.2.1

Multiply $2$ by $8$ .

$x \frac{9 \cdot 289}{{(16 + 3)}^{2} \cdot 8}$

Step 16.2.2

Add $16$ and $3$ .

$x \frac{9 \cdot 289}{19^{2} \cdot 8}$

Step 16.2.3

Raise $19$ to the power of $2$ .

$x \frac{9 \cdot 289}{361 \cdot 8}$

Step 16.3

Multiply $9$ by $289$ .

$x \frac{2601}{361 \cdot 8}$

Step 16.4

Multiply $361$ by $8$ .

$x \frac{2601}{2888}$

Step 16.5

Combine $x$ and $\frac{2601}{2888}$ .

$\frac{x \cdot 2601}{2888}$

Step 16.6

Move $2601$ to the left of $x$ .

$\frac{2601 x}{2888}$