Calculus Examples

$lim_{w \to k} \frac{w^{2} + 9 k w + 8 k^{2}}{w^{2} + k w}$

Step 1

Split the limit using the Limits Quotient Rule on the limit as $w$ approaches $k$ .

$\frac{lim_{w \to k} w^{2} + 9 k w + 8 k^{2}}{lim_{w \to k} w^{2} + k w}$

Step 2

Split the limit using the Sum of Limits Rule on the limit as $w$ approaches $k$ .

$\frac{lim_{w \to k} w^{2} + lim_{w \to k} 9 k w + lim_{w \to k} 8 k^{2}}{lim_{w \to k} w^{2} + k w}$

Step 3

Move the exponent $2$ from $w^{2}$ outside the limit using the Limits Power Rule.

$\frac{{(lim_{w \to k} w)}^{2} + lim_{w \to k} 9 k w + lim_{w \to k} 8 k^{2}}{lim_{w \to k} w^{2} + k w}$

Step 4

Move the term $9 k$ outside of the limit because it is constant with respect to $w$ .

$\frac{{(lim_{w \to k} w)}^{2} + 9 k lim_{w \to k} w + lim_{w \to k} 8 k^{2}}{lim_{w \to k} w^{2} + k w}$

Step 5

Evaluate the limit of $8 k^{2}$ which is constant as $w$ approaches $k$ .

$\frac{{(lim_{w \to k} w)}^{2} + 9 k lim_{w \to k} w + 8 k^{2}}{lim_{w \to k} w^{2} + k w}$

Step 6

Split the limit using the Sum of Limits Rule on the limit as $w$ approaches $k$ .

$\frac{{(lim_{w \to k} w)}^{2} + 9 k lim_{w \to k} w + 8 k^{2}}{lim_{w \to k} w^{2} + lim_{w \to k} k w}$

Step 7

Move the exponent $2$ from $w^{2}$ outside the limit using the Limits Power Rule.

$\frac{{(lim_{w \to k} w)}^{2} + 9 k lim_{w \to k} w + 8 k^{2}}{{(lim_{w \to k} w)}^{2} + lim_{w \to k} k w}$

Step 8

Move the term $k$ outside of the limit because it is constant with respect to $w$ .

$\frac{{(lim_{w \to k} w)}^{2} + 9 k lim_{w \to k} w + 8 k^{2}}{{(lim_{w \to k} w)}^{2} + k lim_{w \to k} w}$

Step 9

Evaluate the limits by plugging in $k$ for all occurrences of $w$ .

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Step 9.1

Evaluate the limit of $w$ by plugging in $k$ for $w$ .

$\frac{k^{2} + 9 k lim_{w \to k} w + 8 k^{2}}{{(lim_{w \to k} w)}^{2} + k lim_{w \to k} w}$

Step 9.2

Evaluate the limit of $w$ by plugging in $k$ for $w$ .

$\frac{k^{2} + 9 k \cdot k + 8 k^{2}}{{(lim_{w \to k} w)}^{2} + k lim_{w \to k} w}$

Step 9.3

Evaluate the limit of $w$ by plugging in $k$ for $w$ .

$\frac{k^{2} + 9 k \cdot k + 8 k^{2}}{k^{2} + k lim_{w \to k} w}$

Step 9.4

Evaluate the limit of $w$ by plugging in $k$ for $w$ .

$\frac{k^{2} + 9 k \cdot k + 8 k^{2}}{k^{2} + k \cdot k}$

Step 10

Simplify the answer.

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Step 10.1

Simplify the numerator.

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Step 10.1.1

Add $k^{2}$ and $8 k^{2}$ .

$\frac{9 k^{2} + 9 k \cdot k}{k^{2} + k \cdot k}$

Step 10.1.2

Factor $9 k$ out of $9 k^{2} + 9 k \cdot k$ .

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Step 10.1.2.1

Factor $9 k$ out of $9 k^{2}$ .

$\frac{9 k (k) + 9 k \cdot k}{k^{2} + k \cdot k}$

Step 10.1.2.2

Factor $9 k$ out of $9 k \cdot k$ .

$\frac{9 k (k) + 9 k (k)}{k^{2} + k \cdot k}$

Step 10.1.2.3

Factor $9 k$ out of $9 k (k) + 9 k (k)$ .

$\frac{9 k (k + k)}{k^{2} + k \cdot k}$

Step 10.1.3

Add $k$ and $k$ .

$\frac{9 k \cdot 2 k}{k^{2} + k \cdot k}$

Step 10.1.4

Combine exponents.

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Step 10.1.4.1

Multiply $2$ by $9$ .

$\frac{18 k \cdot k}{k^{2} + k \cdot k}$

Step 10.1.4.2

Raise $k$ to the power of $1$ .

$\frac{18 (k^{1} k)}{k^{2} + k \cdot k}$

Step 10.1.4.3

Raise $k$ to the power of $1$ .

$\frac{18 (k^{1} k^{1})}{k^{2} + k \cdot k}$

Step 10.1.4.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{18 k^{1 + 1}}{k^{2} + k \cdot k}$

Step 10.1.4.5

Add $1$ and $1$ .

$\frac{18 k^{2}}{k^{2} + k \cdot k}$

Step 10.2

Simplify the denominator.

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Step 10.2.1

Factor $k$ out of $k^{2} + k \cdot k$ .

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Step 10.2.1.1

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

$\frac{18 k^{2}}{k \cdot k + k \cdot k}$

Step 10.2.1.2

Factor $k$ out of $k \cdot k$ .

$\frac{18 k^{2}}{k \cdot k + k (k)}$

Step 10.2.1.3

Factor $k$ out of $k \cdot k + k (k)$ .

$\frac{18 k^{2}}{k (k + k)}$

Step 10.2.2

Add $k$ and $k$ .

$\frac{18 k^{2}}{k \cdot 2 k}$

Step 10.2.3

Combine exponents.

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Step 10.2.3.1

Raise $k$ to the power of $1$ .

$\frac{18 k^{2}}{2 (k^{1} k)}$

Step 10.2.3.2

Raise $k$ to the power of $1$ .

$\frac{18 k^{2}}{2 (k^{1} k^{1})}$

Step 10.2.3.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{18 k^{2}}{2 k^{1 + 1}}$

Step 10.2.3.4

Add $1$ and $1$ .

$\frac{18 k^{2}}{2 k^{2}}$

Step 10.3

Cancel the common factor of $18$ and $2$ .

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Step 10.3.1

Factor $2$ out of $18 k^{2}$ .

$\frac{2 (9 k^{2})}{2 k^{2}}$

Step 10.3.2

Cancel the common factors.

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Step 10.3.2.1

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

$\frac{2 (9 k^{2})}{2 (k^{2})}$

Step 10.3.2.2

Cancel the common factor.

$\frac{2 (9 k^{2})}{2 k^{2}}$

Step 10.3.2.3

Rewrite the expression.

$\frac{9 k^{2}}{k^{2}}$

Step 10.4

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

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Step 10.4.1

Cancel the common factor.

$\frac{9 k^{2}}{k^{2}}$

Step 10.4.2

Divide $9$ by $1$ .

$9$