Calculus Examples

$lim_{n \to 8} \frac{{(n + 3)}^{2}}{7 n + 17}$

Step 1

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

$\frac{lim_{n \to 8} {(n + 3)}^{2}}{lim_{n \to 8} 7 n + 17}$

Step 2

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

$\frac{{(lim_{n \to 8} n + 3)}^{2}}{lim_{n \to 8} 7 n + 17}$

Step 3

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

$\frac{{(lim_{n \to 8} n + lim_{n \to 8} 3)}^{2}}{lim_{n \to 8} 7 n + 17}$

Step 4

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

$\frac{{(lim_{n \to 8} n + 3)}^{2}}{lim_{n \to 8} 7 n + 17}$

Step 5

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

$\frac{{(lim_{n \to 8} n + 3)}^{2}}{lim_{n \to 8} 7 n + lim_{n \to 8} 17}$

Step 6

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

$\frac{{(lim_{n \to 8} n + 3)}^{2}}{7 lim_{n \to 8} n + lim_{n \to 8} 17}$

Step 7

Evaluate the limit of $17$ which is constant as $n$ approaches $8$ .

$\frac{{(lim_{n \to 8} n + 3)}^{2}}{7 lim_{n \to 8} n + 17}$

Step 8

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

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Step 8.1

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

$\frac{{(8 + 3)}^{2}}{7 lim_{n \to 8} n + 17}$

Step 8.2

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

$\frac{{(8 + 3)}^{2}}{7 \cdot 8 + 17}$

Step 9

Simplify the answer.

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Step 9.1

Simplify the numerator.

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Step 9.1.1

Add $8$ and $3$ .

$\frac{11^{2}}{7 \cdot 8 + 17}$

Step 9.1.2

Raise $11$ to the power of $2$ .

$\frac{121}{7 \cdot 8 + 17}$

Step 9.2

Simplify the denominator.

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Step 9.2.1

Multiply $7$ by $8$ .

$\frac{121}{56 + 17}$

Step 9.2.2

Add $56$ and $17$ .

$\frac{121}{73}$