Calculus Examples

$lim_{r \to 0^{+}} - 11 (\frac{r}{3}) \ln (\frac{r}{3})$

Step 1

Move the term $- 11$ outside of the limit because it is constant with respect to $r$ .

$- 11 lim_{r \to 0^{+}} \frac{r}{3} \ln (\frac{r}{3})$

Step 2

Combine $\frac{r}{3}$ and $\ln (\frac{r}{3})$ .

$- 11 lim_{r \to 0^{+}} \frac{r \ln (\frac{r}{3})}{3}$

Step 3

Move the term $\frac{1}{3}$ outside of the limit because it is constant with respect to $r$ .

$- 11 (\frac{1}{3}) lim_{r \to 0^{+}} r \ln (\frac{r}{3})$

Step 4

Rewrite $r \ln (\frac{r}{3})$ as $\frac{\ln (\frac{r}{3})}{\frac{1}{r}}$ .

$- 11 (\frac{1}{3}) lim_{r \to 0^{+}} \frac{\ln (\frac{r}{3})}{\frac{1}{r}}$

Step 5

Apply L'Hospital's rule.

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Step 5.1

Evaluate the limit of the numerator and the limit of the denominator.

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Step 5.1.1

Take the limit of the numerator and the limit of the denominator.

$- 11 (\frac{1}{3}) \frac{lim_{r \to 0^{+}} \ln (\frac{r}{3})}{lim_{r \to 0^{+}} \frac{1}{r}}$

Step 5.1.2

As $r$ approaches $0$ from the right side, $\ln (\frac{r}{3})$ decreases without bound.

$- 11 (\frac{1}{3}) \frac{- \infty}{lim_{r \to 0^{+}} \frac{1}{r}}$

Step 5.1.3

Since the numerator is a constant and the denominator approaches $0$ when $r$ approaches $0$ from the right, the fraction $\frac{1}{r}$ approaches infinity.

$- 11 (\frac{1}{3}) \frac{- \infty}{\infty}$

Step 5.1.4

Infinity divided by infinity is undefined.

Undefined

$- 11 (\frac{1}{3}) \frac{- \infty}{\infty}$

Step 5.2

Since $\frac{- \infty}{\infty}$ is of indeterminate form, apply L'Hospital's Rule. L'Hospital's Rule states that the limit of a quotient of functions is equal to the limit of the quotient of their derivatives.

$lim_{r \to 0^{+}} \frac{\ln (\frac{r}{3})}{\frac{1}{r}} = lim_{r \to 0^{+}} \frac{\frac{d}{d r} [\ln (\frac{r}{3})]}{\frac{d}{d r} [\frac{1}{r}]}$

Step 5.3

Find the derivative of the numerator and denominator.

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Step 5.3.1

Differentiate the numerator and denominator.

$- 11 (\frac{1}{3}) lim_{r \to 0^{+}} \frac{\frac{d}{d r} [\ln (\frac{r}{3})]}{\frac{d}{d r} [\frac{1}{r}]}$

Step 5.3.2

Differentiate using the chain rule, which states that $\frac{d}{d r} [f (g (r))]$ is $f' (g (r)) g' (r)$ where $f (r) = \ln (r)$ and $g (r) = \frac{r}{3}$ .

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Step 5.3.2.1

To apply the Chain Rule, set $u$ as $\frac{r}{3}$ .

$- 11 (\frac{1}{3}) lim_{r \to 0^{+}} \frac{\frac{d}{d u} [\ln (u)] \frac{d}{d r} [\frac{r}{3}]}{\frac{d}{d r} [\frac{1}{r}]}$

Step 5.3.2.2

The derivative of $\ln (u)$ with respect to $u$ is $\frac{1}{u}$ .

$- 11 (\frac{1}{3}) lim_{r \to 0^{+}} \frac{\frac{1}{u} \frac{d}{d r} [\frac{r}{3}]}{\frac{d}{d r} [\frac{1}{r}]}$

Step 5.3.2.3

Replace all occurrences of $u$ with $\frac{r}{3}$ .

$- 11 (\frac{1}{3}) lim_{r \to 0^{+}} \frac{\frac{1}{\frac{r}{3}} \frac{d}{d r} [\frac{r}{3}]}{\frac{d}{d r} [\frac{1}{r}]}$

Step 5.3.3

Multiply by the reciprocal of the fraction to divide by $\frac{r}{3}$ .

$- 11 (\frac{1}{3}) lim_{r \to 0^{+}} \frac{1 \frac{3}{r} \frac{d}{d r} [\frac{r}{3}]}{\frac{d}{d r} [\frac{1}{r}]}$

Step 5.3.4

Multiply $\frac{3}{r}$ by $1$ .

$- 11 (\frac{1}{3}) lim_{r \to 0^{+}} \frac{\frac{3}{r} \frac{d}{d r} [\frac{r}{3}]}{\frac{d}{d r} [\frac{1}{r}]}$

Step 5.3.5

Since $\frac{1}{3}$ is constant with respect to $r$ , the derivative of $\frac{r}{3}$ with respect to $r$ is $\frac{1}{3} \frac{d}{d r} [r]$ .

$- 11 (\frac{1}{3}) lim_{r \to 0^{+}} \frac{\frac{3}{r} \cdot \frac{1}{3} \frac{d}{d r} [r]}{\frac{d}{d r} [\frac{1}{r}]}$

Step 5.3.6

Multiply $\frac{1}{3}$ by $\frac{3}{r}$ .

$- 11 (\frac{1}{3}) lim_{r \to 0^{+}} \frac{\frac{3}{3 r} \frac{d}{d r} [r]}{\frac{d}{d r} [\frac{1}{r}]}$

Step 5.3.7

Cancel the common factor of $3$ .

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Step 5.3.7.1

Cancel the common factor.

$- 11 (\frac{1}{3}) lim_{r \to 0^{+}} \frac{\frac{3}{3 r} \frac{d}{d r} [r]}{\frac{d}{d r} [\frac{1}{r}]}$

Step 5.3.7.2

Rewrite the expression.

$- 11 (\frac{1}{3}) lim_{r \to 0^{+}} \frac{\frac{1}{r} \frac{d}{d r} [r]}{\frac{d}{d r} [\frac{1}{r}]}$

Step 5.3.8

Differentiate using the Power Rule which states that $\frac{d}{d r} [r^{n}]$ is $n r^{n - 1}$ where $n = 1$ .

$- 11 (\frac{1}{3}) lim_{r \to 0^{+}} \frac{\frac{1}{r} \cdot 1}{\frac{d}{d r} [\frac{1}{r}]}$

Step 5.3.9

Multiply $\frac{1}{r}$ by $1$ .

$- 11 (\frac{1}{3}) lim_{r \to 0^{+}} \frac{\frac{1}{r}}{\frac{d}{d r} [\frac{1}{r}]}$

Step 5.3.10

Rewrite $\frac{1}{r}$ as $r^{- 1}$ .

$- 11 (\frac{1}{3}) lim_{r \to 0^{+}} \frac{\frac{1}{r}}{\frac{d}{d r} [r^{- 1}]}$

Step 5.3.11

Differentiate using the Power Rule which states that $\frac{d}{d r} [r^{n}]$ is $n r^{n - 1}$ where $n = - 1$ .

$- 11 (\frac{1}{3}) lim_{r \to 0^{+}} \frac{\frac{1}{r}}{- r^{- 2}}$

Step 5.3.12

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- 11 (\frac{1}{3}) lim_{r \to 0^{+}} \frac{\frac{1}{r}}{- \frac{1}{r^{2}}}$

Step 5.4

Multiply the numerator by the reciprocal of the denominator.

$- 11 (\frac{1}{3}) lim_{r \to 0^{+}} \frac{1}{r} \cdot - 1 r^{2}$

Step 5.5

Combine $\frac{1}{r}$ and $r^{2}$ .

$- 11 (\frac{1}{3}) lim_{r \to 0^{+}} - \frac{r^{2}}{r}$

Step 5.6

Cancel the common factor of $r^{2}$ and $r$ .

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Step 5.6.1

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

$- 11 (\frac{1}{3}) lim_{r \to 0^{+}} - \frac{r \cdot r}{r}$

Step 5.6.2

Cancel the common factors.

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Step 5.6.2.1

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

$- 11 (\frac{1}{3}) lim_{r \to 0^{+}} - \frac{r \cdot r}{r^{1}}$

Step 5.6.2.2

Factor $r$ out of $r^{1}$ .

$- 11 (\frac{1}{3}) lim_{r \to 0^{+}} - \frac{r \cdot r}{r \cdot 1}$

Step 5.6.2.3

Cancel the common factor.

$- 11 (\frac{1}{3}) lim_{r \to 0^{+}} - \frac{r \cdot r}{r \cdot 1}$

Step 5.6.2.4

Rewrite the expression.

$- 11 (\frac{1}{3}) lim_{r \to 0^{+}} - \frac{r}{1}$

Step 5.6.2.5

Divide $r$ by $1$ .

$- 11 (\frac{1}{3}) lim_{r \to 0^{+}} - r$

Step 6

Evaluate the limit.

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Step 6.1

Evaluate the limit.

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Step 6.1.1

Move the term $- 1$ outside of the limit because it is constant with respect to $r$ .

$- 11 (\frac{1}{3}) \cdot - 1 lim_{r \to 0^{+}} r$

Step 6.1.2

Evaluate the limit of $r$ by plugging in $0$ for $r$ .

$- 11 (\frac{1}{3}) \cdot 0$

Step 6.2

Simplify the answer.

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Step 6.2.1

Combine $- 11$ and $\frac{1}{3}$ .

$\frac{- 11}{3} \cdot 0$

Step 6.2.2

Move the negative in front of the fraction.

$- \frac{11}{3} \cdot 0$

Step 6.2.3

Multiply $- \frac{11}{3} \cdot 0$ .

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Step 6.2.3.1

Multiply $0$ by $- 1$ .

$0 (\frac{11}{3})$

Step 6.2.3.2

Multiply $0$ by $\frac{11}{3}$ .

$0$