Calculus Examples

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

Step 1

Change the two-sided limit into a right sided limit.

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

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.

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

Step 2.1.2

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

$\frac{- \infty}{lim_{x \to 0^{+}} \ln (x)}$

Step 2.1.3

As $x$ approaches $0$ from the right side, $\ln (x)$ decreases without bound.

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

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.

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

Step 2.3

Find the derivative of the numerator and denominator.

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Step 2.3.1

Differentiate the numerator and denominator.

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

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 $g (x) = x^{2} + 2 x$ .

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Step 2.3.2.1

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

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

Step 2.3.2.2

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

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

Step 2.3.2.3

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

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

Step 2.3.3

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

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

Step 2.3.4

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

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

Step 2.3.5

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

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

Step 2.3.6

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

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

Step 2.3.7

Multiply $2$ by $1$ .

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

Step 2.3.8

Simplify.

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Step 2.3.8.1

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

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

Step 2.3.8.2

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

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Step 2.3.8.2.1

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

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

Step 2.3.8.2.2

Factor $x$ out of $2 x$ .

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

Step 2.3.8.2.3

Factor $x$ out of $x \cdot x + x \cdot 2$ .

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

Step 2.3.8.3

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

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

Step 2.3.8.4

Factor $2$ out of $2 x + 2$ .

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Step 2.3.8.4.1

Factor $2$ out of $2 x$ .

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

Step 2.3.8.4.2

Factor $2$ out of $2$ .

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

Step 2.3.8.4.3

Factor $2$ out of $2 (x) + 2 (1)$ .

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

Step 2.3.9

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

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

Step 2.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 2.5

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

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

Step 2.6

Cancel the common factor of $x$ .

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Step 2.6.1

Cancel the common factor.

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

Step 2.6.2

Rewrite the expression.

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

Step 3

Evaluate the limit.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

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

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

Step 3.6

Evaluate the limit of $2$ which is constant as $x$ approaches $0$ .

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

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 $x$ .

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

Step 4.2

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

$2 \frac{0 + 1}{0 + 2}$

Step 5

Simplify the answer.

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Step 5.1

Add $0$ and $1$ .

$2 \frac{1}{0 + 2}$

Step 5.2

Add $0$ and $2$ .

$2 (\frac{1}{2})$

Step 5.3

Cancel the common factor of $2$ .

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Step 5.3.1

Cancel the common factor.

$2 (\frac{1}{2})$

Step 5.3.2

Rewrite the expression.

$1$