Calculus Examples

$lim_{a \to 0} \frac{\ln (1 + a)}{a}$

Step 1

Apply L'Hospital's rule.

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Step 1.1

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

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Step 1.1.1

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

$\frac{lim_{a \to 0} \ln (1 + a)}{lim_{a \to 0} a}$

Step 1.1.2

Evaluate the limit of the numerator.

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Step 1.1.2.1

Evaluate the limit.

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Step 1.1.2.1.1

Move the limit inside the logarithm.

$\frac{\ln (lim_{a \to 0} 1 + a)}{lim_{a \to 0} a}$

Step 1.1.2.1.2

Split the limit using the Sum of Limits Rule on the limit as $a$ approaches $0$ .

$\frac{\ln (lim_{a \to 0} 1 + lim_{a \to 0} a)}{lim_{a \to 0} a}$

Step 1.1.2.1.3

Evaluate the limit of $1$ which is constant as $a$ approaches $0$ .

$\frac{\ln (1 + lim_{a \to 0} a)}{lim_{a \to 0} a}$

Step 1.1.2.2

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

$\frac{\ln (1 + 0)}{lim_{a \to 0} a}$

Step 1.1.2.3

Simplify the answer.

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Step 1.1.2.3.1

Add $1$ and $0$ .

$\frac{\ln (1)}{lim_{a \to 0} a}$

Step 1.1.2.3.2

The natural logarithm of $1$ is $0$ .

$\frac{0}{lim_{a \to 0} a}$

Step 1.1.3

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

$\frac{0}{0}$

Step 1.1.4

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Step 1.2

Since $\frac{0}{0}$ 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.

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

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

$lim_{a \to 0} \frac{\frac{d}{d a} [\ln (1 + a)]}{\frac{d}{d a} [a]}$

Step 1.3.2

Differentiate using the chain rule, which states that $\frac{d}{d a} [f (g (a))]$ is $f' (g (a)) g' (a)$ where $f (a) = \ln (a)$ and $g (a) = 1 + a$ .

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Step 1.3.2.1

To apply the Chain Rule, set $u$ as $1 + a$ .

$lim_{a \to 0} \frac{\frac{d}{d u} [\ln (u)] \frac{d}{d a} [1 + a]}{\frac{d}{d a} [a]}$

Step 1.3.2.2

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

$lim_{a \to 0} \frac{\frac{1}{u} \frac{d}{d a} [1 + a]}{\frac{d}{d a} [a]}$

Step 1.3.2.3

Replace all occurrences of $u$ with $1 + a$ .

$lim_{a \to 0} \frac{\frac{1}{1 + a} \frac{d}{d a} [1 + a]}{\frac{d}{d a} [a]}$

Step 1.3.3

By the Sum Rule, the derivative of $1 + a$ with respect to $a$ is $\frac{d}{d a} [1] + \frac{d}{d a} [a]$ .

$lim_{a \to 0} \frac{\frac{1}{1 + a} (\frac{d}{d a} [1] + \frac{d}{d a} [a])}{\frac{d}{d a} [a]}$

Step 1.3.4

Since $1$ is constant with respect to $a$ , the derivative of $1$ with respect to $a$ is $0$ .

$lim_{a \to 0} \frac{\frac{1}{1 + a} (0 + \frac{d}{d a} [a])}{\frac{d}{d a} [a]}$

Step 1.3.5

Add $0$ and $\frac{d}{d a} [a]$ .

$lim_{a \to 0} \frac{\frac{1}{1 + a} \frac{d}{d a} [a]}{\frac{d}{d a} [a]}$

Step 1.3.6

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

$lim_{a \to 0} \frac{\frac{1}{1 + a} \cdot 1}{\frac{d}{d a} [a]}$

Step 1.3.7

Multiply $\frac{1}{1 + a}$ by $1$ .

$lim_{a \to 0} \frac{\frac{1}{1 + a}}{\frac{d}{d a} [a]}$

Step 1.3.8

Reorder terms.

$lim_{a \to 0} \frac{\frac{1}{a + 1}}{\frac{d}{d a} [a]}$

Step 1.3.9

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

$lim_{a \to 0} \frac{\frac{1}{a + 1}}{1}$

Step 1.4

Multiply the numerator by the reciprocal of the denominator.

$lim_{a \to 0} \frac{1}{a + 1} \cdot 1$

Step 1.5

Multiply $\frac{1}{a + 1}$ by $1$ .

$lim_{a \to 0} \frac{1}{a + 1}$

Step 2

Evaluate the limit.

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Step 2.1

Split the limit using the Limits Quotient Rule on the limit as $a$ approaches $0$ .

$\frac{lim_{a \to 0} 1}{lim_{a \to 0} a + 1}$

Step 2.2

Evaluate the limit of $1$ which is constant as $a$ approaches $0$ .

$\frac{1}{lim_{a \to 0} a + 1}$

Step 2.3

Split the limit using the Sum of Limits Rule on the limit as $a$ approaches $0$ .

$\frac{1}{lim_{a \to 0} a + lim_{a \to 0} 1}$

Step 2.4

Evaluate the limit of $1$ which is constant as $a$ approaches $0$ .

$\frac{1}{lim_{a \to 0} a + 1}$

Step 3

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

$\frac{1}{0 + 1}$

Step 4

Simplify the answer.

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Step 4.1

Add $0$ and $1$ .

$\frac{1}{1}$

Step 4.2

Divide $1$ by $1$ .

$1$