Calculus Examples

$lim_{x \to 0} \frac{\ln (1 + 3 x^{2})}{\ln (1 + 5 x^{2})}$

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.

$\frac{lim_{x \to 0} \ln (1 + 3 x^{2})}{lim_{x \to 0} \ln (1 + 5 x^{2})}$

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.

$\frac{\ln (lim_{x \to 0} 1 + 3 x^{2})}{lim_{x \to 0} \ln (1 + 5 x^{2})}$

Step 1.1.2.1.2

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

$\frac{\ln (lim_{x \to 0} 1 + lim_{x \to 0} 3 x^{2})}{lim_{x \to 0} \ln (1 + 5 x^{2})}$

Step 1.1.2.1.3

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

$\frac{\ln (1 + lim_{x \to 0} 3 x^{2})}{lim_{x \to 0} \ln (1 + 5 x^{2})}$

Step 1.1.2.1.4

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$\frac{\ln (1 + 3 lim_{x \to 0} x^{2})}{lim_{x \to 0} \ln (1 + 5 x^{2})}$

Step 1.1.2.1.5

Move the exponent $2$ from $x^{2}$ outside the limit using the Limits Power Rule.

$\frac{\ln (1 + 3 {(lim_{x \to 0} x)}^{2})}{lim_{x \to 0} \ln (1 + 5 x^{2})}$

Step 1.1.2.2

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

$\frac{\ln (1 + 3 \cdot 0^{2})}{lim_{x \to 0} \ln (1 + 5 x^{2})}$

Step 1.1.2.3

Simplify the answer.

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Step 1.1.2.3.1

Simplify each term.

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Step 1.1.2.3.1.1

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

$\frac{\ln (1 + 3 \cdot 0)}{lim_{x \to 0} \ln (1 + 5 x^{2})}$

Step 1.1.2.3.1.2

Multiply $3$ by $0$ .

$\frac{\ln (1 + 0)}{lim_{x \to 0} \ln (1 + 5 x^{2})}$

Step 1.1.2.3.2

Add $1$ and $0$ .

$\frac{\ln (1)}{lim_{x \to 0} \ln (1 + 5 x^{2})}$

Step 1.1.2.3.3

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

$\frac{0}{lim_{x \to 0} \ln (1 + 5 x^{2})}$

Step 1.1.3

Evaluate the limit of the denominator.

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Step 1.1.3.1

Evaluate the limit.

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Step 1.1.3.1.1

Move the limit inside the logarithm.

$\frac{0}{\ln (lim_{x \to 0} 1 + 5 x^{2})}$

Step 1.1.3.1.2

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

$\frac{0}{\ln (lim_{x \to 0} 1 + lim_{x \to 0} 5 x^{2})}$

Step 1.1.3.1.3

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

$\frac{0}{\ln (1 + lim_{x \to 0} 5 x^{2})}$

Step 1.1.3.1.4

Move the term $5$ outside of the limit because it is constant with respect to $x$ .

$\frac{0}{\ln (1 + 5 lim_{x \to 0} x^{2})}$

Step 1.1.3.1.5

Move the exponent $2$ from $x^{2}$ outside the limit using the Limits Power Rule.

$\frac{0}{\ln (1 + 5 {(lim_{x \to 0} x)}^{2})}$

Step 1.1.3.2

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

$\frac{0}{\ln (1 + 5 \cdot 0^{2})}$

Step 1.1.3.3

Simplify the answer.

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Step 1.1.3.3.1

Simplify each term.

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Step 1.1.3.3.1.1

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

$\frac{0}{\ln (1 + 5 \cdot 0)}$

Step 1.1.3.3.1.2

Multiply $5$ by $0$ .

$\frac{0}{\ln (1 + 0)}$

Step 1.1.3.3.2

Add $1$ and $0$ .

$\frac{0}{\ln (1)}$

Step 1.1.3.3.3

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

$\frac{0}{0}$

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.

$lim_{x \to 0} \frac{\ln (1 + 3 x^{2})}{\ln (1 + 5 x^{2})} = lim_{x \to 0} \frac{\frac{d}{d x} [\ln (1 + 3 x^{2})]}{\frac{d}{d x} [\ln (1 + 5 x^{2})]}$

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

$lim_{x \to 0} \frac{\frac{d}{d x} [\ln (1 + 3 x^{2})]}{\frac{d}{d x} [\ln (1 + 5 x^{2})]}$

Step 1.3.2

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 $g (x) = 1 + 3 x^{2}$ .

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Step 1.3.2.1

To apply the Chain Rule, set $u$ as $1 + 3 x^{2}$ .

$lim_{x \to 0} \frac{\frac{d}{d u} [\ln (u)] \frac{d}{d x} [1 + 3 x^{2}]}{\frac{d}{d x} [\ln (1 + 5 x^{2})]}$

Step 1.3.2.2

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

$lim_{x \to 0} \frac{\frac{1}{u} \frac{d}{d x} [1 + 3 x^{2}]}{\frac{d}{d x} [\ln (1 + 5 x^{2})]}$

Step 1.3.2.3

Replace all occurrences of $u$ with $1 + 3 x^{2}$ .

$lim_{x \to 0} \frac{\frac{1}{1 + 3 x^{2}} \frac{d}{d x} [1 + 3 x^{2}]}{\frac{d}{d x} [\ln (1 + 5 x^{2})]}$

Step 1.3.3

By the Sum Rule, the derivative of $1 + 3 x^{2}$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [3 x^{2}]$ .

$lim_{x \to 0} \frac{\frac{1}{1 + 3 x^{2}} (\frac{d}{d x} [1] + \frac{d}{d x} [3 x^{2}])}{\frac{d}{d x} [\ln (1 + 5 x^{2})]}$

Step 1.3.4

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

$lim_{x \to 0} \frac{\frac{1}{1 + 3 x^{2}} (0 + \frac{d}{d x} [3 x^{2}])}{\frac{d}{d x} [\ln (1 + 5 x^{2})]}$

Step 1.3.5

Add $0$ and $\frac{d}{d x} [3 x^{2}]$ .

$lim_{x \to 0} \frac{\frac{1}{1 + 3 x^{2}} \frac{d}{d x} [3 x^{2}]}{\frac{d}{d x} [\ln (1 + 5 x^{2})]}$

Step 1.3.6

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{2}$ with respect to $x$ is $3 \frac{d}{d x} [x^{2}]$ .

$lim_{x \to 0} \frac{\frac{1}{1 + 3 x^{2}} (3 \frac{d}{d x} [x^{2}])}{\frac{d}{d x} [\ln (1 + 5 x^{2})]}$

Step 1.3.7

Combine $3$ and $\frac{1}{1 + 3 x^{2}}$ .

$lim_{x \to 0} \frac{\frac{3}{1 + 3 x^{2}} \frac{d}{d x} [x^{2}]}{\frac{d}{d x} [\ln (1 + 5 x^{2})]}$

Step 1.3.8

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

$lim_{x \to 0} \frac{\frac{3}{1 + 3 x^{2}} (2 x)}{\frac{d}{d x} [\ln (1 + 5 x^{2})]}$

Step 1.3.9

Combine $2$ and $\frac{3}{1 + 3 x^{2}}$ .

$lim_{x \to 0} \frac{\frac{2 \cdot 3}{1 + 3 x^{2}} x}{\frac{d}{d x} [\ln (1 + 5 x^{2})]}$

Step 1.3.10

Multiply $2$ by $3$ .

$lim_{x \to 0} \frac{\frac{6}{1 + 3 x^{2}} x}{\frac{d}{d x} [\ln (1 + 5 x^{2})]}$

Step 1.3.11

Combine $\frac{6}{1 + 3 x^{2}}$ and $x$ .

$lim_{x \to 0} \frac{\frac{6 x}{1 + 3 x^{2}}}{\frac{d}{d x} [\ln (1 + 5 x^{2})]}$

Step 1.3.12

Reorder terms.

$lim_{x \to 0} \frac{\frac{6 x}{3 x^{2} + 1}}{\frac{d}{d x} [\ln (1 + 5 x^{2})]}$

Step 1.3.13

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 $g (x) = 1 + 5 x^{2}$ .

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Step 1.3.13.1

To apply the Chain Rule, set $u$ as $1 + 5 x^{2}$ .

$lim_{x \to 0} \frac{\frac{6 x}{3 x^{2} + 1}}{\frac{d}{d u} [\ln (u)] \frac{d}{d x} [1 + 5 x^{2}]}$

Step 1.3.13.2

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

$lim_{x \to 0} \frac{\frac{6 x}{3 x^{2} + 1}}{\frac{1}{u} \frac{d}{d x} [1 + 5 x^{2}]}$

Step 1.3.13.3

Replace all occurrences of $u$ with $1 + 5 x^{2}$ .

$lim_{x \to 0} \frac{\frac{6 x}{3 x^{2} + 1}}{\frac{1}{1 + 5 x^{2}} \frac{d}{d x} [1 + 5 x^{2}]}$

Step 1.3.14

By the Sum Rule, the derivative of $1 + 5 x^{2}$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [5 x^{2}]$ .

$lim_{x \to 0} \frac{\frac{6 x}{3 x^{2} + 1}}{\frac{1}{1 + 5 x^{2}} (\frac{d}{d x} [1] + \frac{d}{d x} [5 x^{2}])}$

Step 1.3.15

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

$lim_{x \to 0} \frac{\frac{6 x}{3 x^{2} + 1}}{\frac{1}{1 + 5 x^{2}} (0 + \frac{d}{d x} [5 x^{2}])}$

Step 1.3.16

Add $0$ and $\frac{d}{d x} [5 x^{2}]$ .

$lim_{x \to 0} \frac{\frac{6 x}{3 x^{2} + 1}}{\frac{1}{1 + 5 x^{2}} \frac{d}{d x} [5 x^{2}]}$

Step 1.3.17

Since $5$ is constant with respect to $x$ , the derivative of $5 x^{2}$ with respect to $x$ is $5 \frac{d}{d x} [x^{2}]$ .

$lim_{x \to 0} \frac{\frac{6 x}{3 x^{2} + 1}}{\frac{1}{1 + 5 x^{2}} (5 \frac{d}{d x} [x^{2}])}$

Step 1.3.18

Combine $5$ and $\frac{1}{1 + 5 x^{2}}$ .

$lim_{x \to 0} \frac{\frac{6 x}{3 x^{2} + 1}}{\frac{5}{1 + 5 x^{2}} \frac{d}{d x} [x^{2}]}$

Step 1.3.19

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

$lim_{x \to 0} \frac{\frac{6 x}{3 x^{2} + 1}}{\frac{5}{1 + 5 x^{2}} (2 x)}$

Step 1.3.20

Combine $2$ and $\frac{5}{1 + 5 x^{2}}$ .

$lim_{x \to 0} \frac{\frac{6 x}{3 x^{2} + 1}}{\frac{2 \cdot 5}{1 + 5 x^{2}} x}$

Step 1.3.21

Multiply $2$ by $5$ .

$lim_{x \to 0} \frac{\frac{6 x}{3 x^{2} + 1}}{\frac{10}{1 + 5 x^{2}} x}$

Step 1.3.22

Combine $\frac{10}{1 + 5 x^{2}}$ and $x$ .

$lim_{x \to 0} \frac{\frac{6 x}{3 x^{2} + 1}}{\frac{10 x}{1 + 5 x^{2}}}$

Step 1.3.23

Reorder terms.

$lim_{x \to 0} \frac{\frac{6 x}{3 x^{2} + 1}}{\frac{10 x}{5 x^{2} + 1}}$

Step 1.4

Multiply the numerator by the reciprocal of the denominator.

$lim_{x \to 0} \frac{6 x}{3 x^{2} + 1} \cdot \frac{5 x^{2} + 1}{10 x}$

Step 1.5

Multiply $\frac{6 x}{3 x^{2} + 1}$ by $\frac{5 x^{2} + 1}{10 x}$ .

$lim_{x \to 0} \frac{6 x (5 x^{2} + 1)}{(3 x^{2} + 1) (10 x)}$

Step 1.6

Reduce.

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Step 1.6.1

Cancel the common factor of $6$ and $10$ .

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Step 1.6.1.1

Factor $2$ out of $6 x (5 x^{2} + 1)$ .

$lim_{x \to 0} \frac{2 (3 x (5 x^{2} + 1))}{(3 x^{2} + 1) (10 x)}$

Step 1.6.1.2

Cancel the common factors.

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Step 1.6.1.2.1

Factor $2$ out of $(3 x^{2} + 1) (10 x)$ .

$lim_{x \to 0} \frac{2 (3 x (5 x^{2} + 1))}{2 ((3 x^{2} + 1) (5 x))}$

Step 1.6.1.2.2

Cancel the common factor.

$lim_{x \to 0} \frac{2 (3 x (5 x^{2} + 1))}{2 ((3 x^{2} + 1) (5 x))}$

Step 1.6.1.2.3

Rewrite the expression.

$lim_{x \to 0} \frac{3 x (5 x^{2} + 1)}{(3 x^{2} + 1) (5 x)}$

Step 1.6.2

Cancel the common factor of $x$ .

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Step 1.6.2.1

Cancel the common factor.

$lim_{x \to 0} \frac{3 x (5 x^{2} + 1)}{(3 x^{2} + 1) (5 x)}$

Step 1.6.2.2

Rewrite the expression.

$lim_{x \to 0} \frac{3 (5 x^{2} + 1)}{(3 x^{2} + 1) \cdot (5)}$

Step 2

Evaluate the limit.

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Step 2.1

Move the term $\frac{3}{5}$ outside of the limit because it is constant with respect to $x$ .

$\frac{3}{5} lim_{x \to 0} \frac{5 x^{2} + 1}{3 x^{2} + 1}$

Step 2.2

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

$\frac{3}{5} \cdot \frac{lim_{x \to 0} 5 x^{2} + 1}{lim_{x \to 0} 3 x^{2} + 1}$

Step 2.3

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

$\frac{3}{5} \cdot \frac{lim_{x \to 0} 5 x^{2} + lim_{x \to 0} 1}{lim_{x \to 0} 3 x^{2} + 1}$

Step 2.4

Move the term $5$ outside of the limit because it is constant with respect to $x$ .

$\frac{3}{5} \cdot \frac{5 lim_{x \to 0} x^{2} + lim_{x \to 0} 1}{lim_{x \to 0} 3 x^{2} + 1}$

Step 2.5

Move the exponent $2$ from $x^{2}$ outside the limit using the Limits Power Rule.

$\frac{3}{5} \cdot \frac{5 {(lim_{x \to 0} x)}^{2} + lim_{x \to 0} 1}{lim_{x \to 0} 3 x^{2} + 1}$

Step 2.6

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

$\frac{3}{5} \cdot \frac{5 {(lim_{x \to 0} x)}^{2} + 1}{lim_{x \to 0} 3 x^{2} + 1}$

Step 2.7

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

$\frac{3}{5} \cdot \frac{5 {(lim_{x \to 0} x)}^{2} + 1}{lim_{x \to 0} 3 x^{2} + lim_{x \to 0} 1}$

Step 2.8

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$\frac{3}{5} \cdot \frac{5 {(lim_{x \to 0} x)}^{2} + 1}{3 lim_{x \to 0} x^{2} + lim_{x \to 0} 1}$

Step 2.9

Move the exponent $2$ from $x^{2}$ outside the limit using the Limits Power Rule.

$\frac{3}{5} \cdot \frac{5 {(lim_{x \to 0} x)}^{2} + 1}{3 {(lim_{x \to 0} x)}^{2} + lim_{x \to 0} 1}$

Step 2.10

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

$\frac{3}{5} \cdot \frac{5 {(lim_{x \to 0} x)}^{2} + 1}{3 {(lim_{x \to 0} x)}^{2} + 1}$

Step 3

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

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Step 3.1

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

$\frac{3}{5} \cdot \frac{5 \cdot 0^{2} + 1}{3 {(lim_{x \to 0} x)}^{2} + 1}$

Step 3.2

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

$\frac{3}{5} \cdot \frac{5 \cdot 0^{2} + 1}{3 \cdot 0^{2} + 1}$

Step 4

Simplify the answer.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

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

$\frac{3}{5} \cdot \frac{5 \cdot 0 + 1}{3 \cdot 0^{2} + 1}$

Step 4.1.2

Multiply $5$ by $0$ .

$\frac{3}{5} \cdot \frac{0 + 1}{3 \cdot 0^{2} + 1}$

Step 4.1.3

Add $0$ and $1$ .

$\frac{3}{5} \cdot \frac{1}{3 \cdot 0^{2} + 1}$

Step 4.2

Simplify the denominator.

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Step 4.2.1

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

$\frac{3}{5} \cdot \frac{1}{3 \cdot 0 + 1}$

Step 4.2.2

Multiply $3$ by $0$ .

$\frac{3}{5} \cdot \frac{1}{0 + 1}$

Step 4.2.3

Add $0$ and $1$ .

$\frac{3}{5} \cdot \frac{1}{1}$

Step 4.3

Cancel the common factor of $1$ .

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Step 4.3.1

Cancel the common factor.

$\frac{3}{5} \cdot \frac{1}{1}$

Step 4.3.2

Rewrite the expression.

$\frac{3}{5} \cdot 1$

Step 4.4

Multiply $\frac{3}{5}$ by $1$ .

$\frac{3}{5}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{3}{5}$

Decimal Form:

$0.6$