Calculus Examples

$\frac{lim_{q \to 2} 4 q^{2} - q + 2}{8}$

Step 1

Split the limit using the Sum of Limits Rule on the limit as $q$ approaches $2$ .

$\frac{lim_{q \to 2} 4 q^{2} - lim_{q \to 2} q + lim_{q \to 2} 2}{8}$

Step 2

Move the term $4$ outside of the limit because it is constant with respect to $q$ .

$\frac{4 lim_{q \to 2} q^{2} - lim_{q \to 2} q + lim_{q \to 2} 2}{8}$

Step 3

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

$\frac{4 {(lim_{q \to 2} q)}^{2} - lim_{q \to 2} q + lim_{q \to 2} 2}{8}$

Step 4

Evaluate the limit of $2$ which is constant as $q$ approaches $2$ .

$\frac{4 {(lim_{q \to 2} q)}^{2} - lim_{q \to 2} q + 2}{8}$

Step 5

Evaluate the limits by plugging in $2$ for all occurrences of $q$ .

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Step 5.1

Evaluate the limit of $q$ by plugging in $2$ for $q$ .

$\frac{4 \cdot 2^{2} - lim_{q \to 2} q + 2}{8}$

Step 5.2

Evaluate the limit of $q$ by plugging in $2$ for $q$ .

$\frac{4 \cdot 2^{2} - 2 + 2}{8}$

Step 6

Simplify the answer.

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Step 6.1

Cancel the common factor of $4 \cdot 2^{2} - 2 + 2$ and $8$ .

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Step 6.1.1

Factor $2$ out of $4 \cdot 2^{2}$ .

$\frac{2 (2 \cdot 2^{2}) - 2 + 2}{8}$

Step 6.1.2

Factor $2$ out of $- 2$ .

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

Step 6.1.3

Factor $2$ out of $2 (2 \cdot 2^{2}) + 2 (- 1)$ .

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

Step 6.1.4

Factor $2$ out of $2$ .

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

Step 6.1.5

Factor $2$ out of $2 (2 \cdot 2^{2} - 1) + 2 (1)$ .

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

Step 6.1.6

Cancel the common factors.

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Step 6.1.6.1

Factor $2$ out of $8$ .

$\frac{2 (2 \cdot 2^{2} - 1 + 1)}{2 (4)}$

Step 6.1.6.2

Cancel the common factor.

$\frac{2 (2 \cdot 2^{2} - 1 + 1)}{2 \cdot 4}$

Step 6.1.6.3

Rewrite the expression.

$\frac{2 \cdot 2^{2} - 1 + 1}{4}$

Step 6.2

Simplify the numerator.

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Step 6.2.1

Multiply $2$ by $2^{2}$ by adding the exponents.

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Step 6.2.1.1

Multiply $2$ by $2^{2}$ .

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Step 6.2.1.1.1

Raise $2$ to the power of $1$ .

$\frac{2^{1} \cdot 2^{2} - 1 + 1}{4}$

Step 6.2.1.1.2

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

$\frac{2^{1 + 2} - 1 + 1}{4}$

Step 6.2.1.2

Add $1$ and $2$ .

$\frac{2^{3} - 1 + 1}{4}$

Step 6.2.2

Raise $2$ to the power of $3$ .

$\frac{8 - 1 + 1}{4}$

Step 6.2.3

Subtract $1$ from $8$ .

$\frac{7 + 1}{4}$

Step 6.2.4

Add $7$ and $1$ .

$\frac{8}{4}$

Step 6.3

Divide $8$ by $4$ .

$2$