Calculus Examples

$lim_{x \to 1} \frac{\ln (x)}{{(x - 1)}^{3}}$

Step 1

Apply L'Hospital's rule.

Tap for more steps...

Step 1.1

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

Tap for more steps...

Step 1.1.1

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

$\frac{lim_{x \to 1} \ln (x)}{lim_{x \to 1} {(x - 1)}^{3}}$

Step 1.1.2

Evaluate the limit of the numerator.

Tap for more steps...

Step 1.1.2.1

Move the limit inside the logarithm.

$\frac{\ln (lim_{x \to 1} x)}{lim_{x \to 1} {(x - 1)}^{3}}$

Step 1.1.2.2

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

$\frac{\ln (1)}{lim_{x \to 1} {(x - 1)}^{3}}$

Step 1.1.2.3

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

$\frac{0}{lim_{x \to 1} {(x - 1)}^{3}}$

Step 1.1.3

Evaluate the limit of the denominator.

Tap for more steps...

Step 1.1.3.1

Evaluate the limit.

Tap for more steps...

Step 1.1.3.1.1

Move the exponent $3$ from ${(x - 1)}^{3}$ outside the limit using the Limits Power Rule.

$\frac{0}{{(lim_{x \to 1} x - 1)}^{3}}$

Step 1.1.3.1.2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $1$ .

$\frac{0}{{(lim_{x \to 1} x - lim_{x \to 1} 1)}^{3}}$

Step 1.1.3.1.3

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

$\frac{0}{{(lim_{x \to 1} x - 1 \cdot 1)}^{3}}$

Step 1.1.3.2

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

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

Step 1.1.3.3

Simplify the answer.

Tap for more steps...

Step 1.1.3.3.1

Multiply $- 1$ by $1$ .

$\frac{0}{{(1 - 1)}^{3}}$

Step 1.1.3.3.2

Subtract $1$ from $1$ .

$\frac{0}{0^{3}}$

Step 1.1.3.3.3

Raising $0$ to any positive power yields $0$ .

$\frac{0}{0}$

Step 1.1.3.3.4

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

Undefined

$\frac{0}{0}$

Step 1.1.3.4

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

Undefined

$\frac{0}{0}$

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.

$lim_{x \to 1} \frac{\ln (x)}{{(x - 1)}^{3}} = lim_{x \to 1} \frac{\frac{d}{d x} [\ln (x)]}{\frac{d}{d x} [{(x - 1)}^{3}]}$

Step 1.3

Find the derivative of the numerator and denominator.

Tap for more steps...

Step 1.3.1

Differentiate the numerator and denominator.

$lim_{x \to 1} \frac{\frac{d}{d x} [\ln (x)]}{\frac{d}{d x} [{(x - 1)}^{3}]}$

Step 1.3.2

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

$lim_{x \to 1} \frac{\frac{1}{x}}{\frac{d}{d x} [{(x - 1)}^{3}]}$

Step 1.3.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{3}$ and $g (x) = x - 1$ .

Tap for more steps...

Step 1.3.3.1

To apply the Chain Rule, set $u$ as $x - 1$ .

$lim_{x \to 1} \frac{\frac{1}{x}}{\frac{d}{d u} [u^{3}] \frac{d}{d x} [x - 1]}$

Step 1.3.3.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 3$ .

$lim_{x \to 1} \frac{\frac{1}{x}}{3 u^{2} \frac{d}{d x} [x - 1]}$

Step 1.3.3.3

Replace all occurrences of $u$ with $x - 1$ .

$lim_{x \to 1} \frac{\frac{1}{x}}{3 {(x - 1)}^{2} \frac{d}{d x} [x - 1]}$

Step 1.3.4

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

$lim_{x \to 1} \frac{\frac{1}{x}}{3 {(x - 1)}^{2} (\frac{d}{d x} [x] + \frac{d}{d x} [- 1])}$

Step 1.3.5

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

$lim_{x \to 1} \frac{\frac{1}{x}}{3 {(x - 1)}^{2} (1 + \frac{d}{d x} [- 1])}$

Step 1.3.6

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

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

Step 1.3.7

Add $1$ and $0$ .

$lim_{x \to 1} \frac{\frac{1}{x}}{3 {(x - 1)}^{2} \cdot 1}$

Step 1.3.8

Multiply $3$ by $1$ .

$lim_{x \to 1} \frac{\frac{1}{x}}{3 {(x - 1)}^{2}}$

Step 1.4

Multiply the numerator by the reciprocal of the denominator.

$lim_{x \to 1} \frac{1}{x} \cdot \frac{1}{3 {(x - 1)}^{2}}$

Step 1.5

Multiply $\frac{1}{x}$ by $\frac{1}{3 {(x - 1)}^{2}}$ .

$lim_{x \to 1} \frac{1}{x (3 {(x - 1)}^{2})}$

Step 2

Since the numerator is positive and the denominator $x (3 {(x - 1)}^{2})$ approaches zero and is greater than zero for $x$ near $1$ on both sides, the function increases without bound.

$\infty$