Calculus Examples

$lim_{x \to 0^{+}} x \ln (x + x^{2})$

Step 1

Rewrite $x \ln (x + x^{2})$ as $\frac{\ln (x + x^{2})}{\frac{1}{x}}$ .

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

Step 2

Apply L'Hospital's rule.

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Step 2.1

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

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Step 2.1.1

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

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

Step 2.1.2

As $x$ approaches $0$ from the right side, $\ln (x + x^{2})$ decreases without bound.

$\frac{- \infty}{lim_{x \to 0^{+}} \frac{1}{x}}$

Step 2.1.3

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

$\frac{- \infty}{\infty}$

Step 2.1.4

Infinity divided by infinity is undefined.

Undefined

$\frac{- \infty}{\infty}$

Step 2.2

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.

$lim_{x \to 0^{+}} \frac{\ln (x + x^{2})}{\frac{1}{x}} = lim_{x \to 0^{+}} \frac{\frac{d}{d x} [\ln (x + x^{2})]}{\frac{d}{d x} [\frac{1}{x}]}$

Step 2.3

Find the derivative of the numerator and denominator.

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Step 2.3.1

Differentiate the numerator and denominator.

$lim_{x \to 0^{+}} \frac{\frac{d}{d x} [\ln (x + x^{2})]}{\frac{d}{d x} [\frac{1}{x}]}$

Step 2.3.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \ln (x)$ and $g (x) = x + x^{2}$ .

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Step 2.3.2.1

To apply the Chain Rule, set $u$ as $x + x^{2}$ .

$lim_{x \to 0^{+}} \frac{\frac{d}{d u} [\ln (u)] \frac{d}{d x} [x + x^{2}]}{\frac{d}{d x} [\frac{1}{x}]}$

Step 2.3.2.2

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

$lim_{x \to 0^{+}} \frac{\frac{1}{u} \frac{d}{d x} [x + x^{2}]}{\frac{d}{d x} [\frac{1}{x}]}$

Step 2.3.2.3

Replace all occurrences of $u$ with $x + x^{2}$ .

$lim_{x \to 0^{+}} \frac{\frac{1}{x + x^{2}} \frac{d}{d x} [x + x^{2}]}{\frac{d}{d x} [\frac{1}{x}]}$

Step 2.3.3

By the Sum Rule, the derivative of $x + x^{2}$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [x^{2}]$ .

$lim_{x \to 0^{+}} \frac{\frac{1}{x + x^{2}} (\frac{d}{d x} [x] + \frac{d}{d x} [x^{2}])}{\frac{d}{d x} [\frac{1}{x}]}$

Step 2.3.4

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

$lim_{x \to 0^{+}} \frac{\frac{1}{x + x^{2}} (1 + \frac{d}{d x} [x^{2}])}{\frac{d}{d x} [\frac{1}{x}]}$

Step 2.3.5

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$lim_{x \to 0^{+}} \frac{\frac{1}{x + x^{2}} (1 + 2 x)}{\frac{d}{d x} [\frac{1}{x}]}$

Step 2.3.6

Simplify.

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Step 2.3.6.1

Reorder the factors of $\frac{1}{x + x^{2}} (1 + 2 x)$ .

$lim_{x \to 0^{+}} \frac{(1 + 2 x) \frac{1}{x + x^{2}}}{\frac{d}{d x} [\frac{1}{x}]}$

Step 2.3.6.2

Factor $x$ out of $x + x^{2}$ .

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Step 2.3.6.2.1

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

$lim_{x \to 0^{+}} \frac{(1 + 2 x) \frac{1}{x^{1} + x^{2}}}{\frac{d}{d x} [\frac{1}{x}]}$

Step 2.3.6.2.2

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

$lim_{x \to 0^{+}} \frac{(1 + 2 x) \frac{1}{x \cdot 1 + x^{2}}}{\frac{d}{d x} [\frac{1}{x}]}$

Step 2.3.6.2.3

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

$lim_{x \to 0^{+}} \frac{(1 + 2 x) \frac{1}{x \cdot 1 + x \cdot x}}{\frac{d}{d x} [\frac{1}{x}]}$

Step 2.3.6.2.4

Factor $x$ out of $x \cdot 1 + x \cdot x$ .

$lim_{x \to 0^{+}} \frac{(1 + 2 x) \frac{1}{x (1 + x)}}{\frac{d}{d x} [\frac{1}{x}]}$

Step 2.3.6.3

Multiply $1 + 2 x$ by $\frac{1}{x (1 + x)}$ .

$lim_{x \to 0^{+}} \frac{\frac{1 + 2 x}{x (1 + x)}}{\frac{d}{d x} [\frac{1}{x}]}$

Step 2.3.7

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

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

Step 2.3.8

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

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

Step 2.3.9

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

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

Step 2.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 2.5

Combine $\frac{1 + 2 x}{x (1 + x)}$ and $x^{2}$ .

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

Step 2.6

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

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Step 2.6.1

Factor $x$ out of $(1 + 2 x) x^{2}$ .

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

Step 2.6.2

Cancel the common factors.

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Step 2.6.2.1

Cancel the common factor.

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

Step 2.6.2.2

Rewrite the expression.

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

Step 2.7

Reorder factors in $- \frac{(1 + 2 x) x}{1 + x}$ .

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

Step 3

Evaluate the limit.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $0$ .

$- \frac{lim_{x \to 0^{+}} x \cdot lim_{x \to 0^{+}} 1 + 2 x}{lim_{x \to 0^{+}} 1 + x}$

Step 3.4

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

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

Step 3.5

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

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

Step 3.6

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

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

Step 3.7

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

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

Step 3.8

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

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

Step 4

Evaluate the limits by plugging in $0$ for all occurrences of $x$ .

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

$- \frac{0 \cdot (1 + 2 \cdot 0)}{1 + 0}$

Step 5

Simplify the answer.

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Step 5.1

Multiply $0$ by $1 + 2 \cdot 0$ .

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

Step 5.2

Add $1$ and $0$ .

$- \frac{0}{1}$

Step 5.3

Divide $0$ by $1$ .

$- 0$

Step 5.4

Multiply $- 1$ by $0$ .

$0$