Calculus Examples

$lim_{r \to 4} (\frac{2}{r} + \frac{1}{2}) (r^{2} - \frac{4}{r})$

Step 1

Split the limit using the Product of Limits Rule on the limit as $r$ approaches $4$ .

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

Step 2

Split the limit using the Sum of Limits Rule on the limit as $r$ approaches $4$ .

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

Step 3

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

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

Step 4

Split the limit using the Limits Quotient Rule on the limit as $r$ approaches $4$ .

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

Step 5

Evaluate the limit of $1$ which is constant as $r$ approaches $4$ .

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

Step 6

Evaluate the limit of $\frac{1}{2}$ which is constant as $r$ approaches $4$ .

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

Step 7

Split the limit using the Sum of Limits Rule on the limit as $r$ approaches $4$ .

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

Step 8

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

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

Step 9

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

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

Step 10

Split the limit using the Limits Quotient Rule on the limit as $r$ approaches $4$ .

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

Step 11

Evaluate the limit of $1$ which is constant as $r$ approaches $4$ .

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

Step 12

Evaluate the limits by plugging in $4$ for all occurrences of $r$ .

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Step 12.1

Evaluate the limit of $r$ by plugging in $4$ for $r$ .

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

Step 12.2

Evaluate the limit of $r$ by plugging in $4$ for $r$ .

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

Step 12.3

Evaluate the limit of $r$ by plugging in $4$ for $r$ .

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

Step 13

Simplify the answer.

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Step 13.1

Cancel the common factor of $2$ .

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Step 13.1.1

Factor $2$ out of $4$ .

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

Step 13.1.2

Cancel the common factor.

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

Step 13.1.3

Rewrite the expression.

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

Step 13.2

Combine the numerators over the common denominator.

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

Step 13.3

Add $1$ and $1$ .

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

Step 13.4

Cancel the common factor of $2$ .

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Step 13.4.1

Cancel the common factor.

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

Step 13.4.2

Rewrite the expression.

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

Step 13.5

Multiply $4^{2} - 4 (\frac{1}{4})$ by $1$ .

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

Step 13.6

Simplify each term.

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Step 13.6.1

Raise $4$ to the power of $2$ .

$16 - 4 (\frac{1}{4})$

Step 13.6.2

Cancel the common factor of $4$ .

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Step 13.6.2.1

Factor $4$ out of $- 4$ .

$16 + 4 (- 1) \frac{1}{4}$

Step 13.6.2.2

Cancel the common factor.

$16 + 4 \cdot - 1 \frac{1}{4}$

Step 13.6.2.3

Rewrite the expression.

$16 - 1$

Step 13.7

Subtract $1$ from $16$ .

$15$