Calculus Examples

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

Step 1

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

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Step 1.1

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

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

Step 1.2

Evaluate the limit of the numerator.

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Step 1.2.1

Split the limit using the Sum of Limits Rule on the limit as

$x$ approaches

$0$ .

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

Step 1.2.2

Move the term

$2$ outside of the limit because it is constant with respect to

$x$ .

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

Step 1.2.3

Move the exponent

$2$ from

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

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

Step 1.2.4

Evaluate the limits by plugging in

$0$ for all occurrences of

$x$ .

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Step 1.2.4.1

Evaluate the limit of

$x$ by plugging in

$0$ for

$x$ .

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

Step 1.2.4.2

Evaluate the limit of

$x$ by plugging in

$0$ for

$x$ .

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

Step 1.2.5

Simplify the answer.

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Step 1.2.5.1

Simplify each term.

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Step 1.2.5.1.1

Raising

$0$ to any positive power yields

$0$ .

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

Step 1.2.5.1.2

Multiply

$2$ by

$0$ .

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

Step 1.2.5.2

Add

$0$ and

$0$ .

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

Step 1.3

Evaluate the limit of the denominator.

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Step 1.3.1

Split the limit using the Sum of Limits Rule on the limit as

$x$ approaches

$0$ .

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

Step 1.3.2

Move the term

$3$ outside of the limit because it is constant with respect to

$x$ .

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

Step 1.3.3

Move the limit inside the logarithm.

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

Step 1.3.4

Split the limit using the Sum of Limits Rule on the limit as

$x$ approaches

$0$ .

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

Step 1.3.5

Evaluate the limit of

$1$ which is constant as

$x$ approaches

$0$ .

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

Step 1.3.6

Move the term

$3$ outside of the limit because it is constant with respect to

$x$ .

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

Step 1.3.7

Evaluate the limits by plugging in

$0$ for all occurrences of

$x$ .

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Step 1.3.7.1

Evaluate the limit of

$x$ by plugging in

$0$ for

$x$ .

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

Step 1.3.7.2

Evaluate the limit of

$x$ by plugging in

$0$ for

$x$ .

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

Step 1.3.8

Simplify the answer.

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Step 1.3.8.1

Simplify each term.

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Step 1.3.8.1.1

Add

$0$ and

$1$ .

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

Step 1.3.8.1.2

The natural logarithm of

$1$ is

$0$ .

$\frac{0}{3 \cdot 0 - 3 \cdot 0}$

Step 1.3.8.1.3

Multiply

$3$ by

$0$ .

$\frac{0}{0 - 3 \cdot 0}$

Step 1.3.8.1.4

Multiply

$- 3$ by

$0$ .

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

Step 1.3.8.2

Add

$0$ and

$0$ .

$\frac{0}{0}$

Step 1.3.8.3

The expression contains a division by

$0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Step 1.3.9

The expression contains a division by

$0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Step 1.4

The expression contains a division by

$0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Step 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.

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

Step 3

Find the derivative of the numerator and denominator.

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Step 3.1

Differentiate the numerator and denominator.

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

Step 3.2

By the Sum Rule, the derivative of

$x + 2 x^{2}$ with respect to

$x$ is

$\frac{d}{d x} [x] + \frac{d}{d x} [2 x^{2}]$ .

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

Step 3.3

Differentiate using the Power Rule which states that

$\frac{d}{d x} [x^{n}]$ is

$n x^{n - 1}$ where

$n = 1$ .

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

Step 3.4

Evaluate

$\frac{d}{d x} [2 x^{2}]$ .

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Step 3.4.1

Since

$2$ is constant with respect to

$x$ , the derivative of

$2 x^{2}$ with respect to

$x$ is

$2 \frac{d}{d x} [x^{2}]$ .

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

Step 3.4.2

Differentiate using the Power Rule which states that

$\frac{d}{d x} [x^{n}]$ is

$n x^{n - 1}$ where

$n = 2$ .

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

Step 3.4.3

Multiply

$2$ by

$2$ .

$lim_{x \to 0} \frac{1 + 4 x}{\frac{d}{d x} [3 \ln (x + 1) - 3 x]}$

Step 3.5

Reorder terms.

$lim_{x \to 0} \frac{4 x + 1}{\frac{d}{d x} [3 \ln (x + 1) - 3 x]}$

Step 3.6

By the Sum Rule, the derivative of

$3 \ln (x + 1) - 3 x$ with respect to

$x$ is

$\frac{d}{d x} [3 \ln (x + 1)] + \frac{d}{d x} [- 3 x]$ .

$lim_{x \to 0} \frac{4 x + 1}{\frac{d}{d x} [3 \ln (x + 1)] + \frac{d}{d x} [- 3 x]}$

Step 3.7

Evaluate

$\frac{d}{d x} [3 \ln (x + 1)]$ .

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Step 3.7.1

Since

$3$ is constant with respect to

$x$ , the derivative of

$3 \ln (x + 1)$ with respect to

$x$ is

$3 \frac{d}{d x} [\ln (x + 1)]$ .

$lim_{x \to 0} \frac{4 x + 1}{3 \frac{d}{d x} [\ln (x + 1)] + \frac{d}{d x} [- 3 x]}$

Step 3.7.2

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

$g (x) = x + 1$ .

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Step 3.7.2.1

To apply the Chain Rule, set

$u$ as

$x + 1$ .

$lim_{x \to 0} \frac{4 x + 1}{3 (\frac{d}{d u} [\ln (u)] \frac{d}{d x} [x + 1]) + \frac{d}{d x} [- 3 x]}$

Step 3.7.2.2

The derivative of

$\ln (u)$ with respect to

$u$ is

$\frac{1}{u}$ .

$lim_{x \to 0} \frac{4 x + 1}{3 (\frac{1}{u} \frac{d}{d x} [x + 1]) + \frac{d}{d x} [- 3 x]}$

Step 3.7.2.3

Replace all occurrences of

$u$ with

$x + 1$ .

$lim_{x \to 0} \frac{4 x + 1}{3 (\frac{1}{x + 1} \frac{d}{d x} [x + 1]) + \frac{d}{d x} [- 3 x]}$

Step 3.7.3

By the Sum Rule, the derivative of

$x + 1$ with respect to

$x$ is

$\frac{d}{d x} [x] + \frac{d}{d x} [1]$ .

$lim_{x \to 0} \frac{4 x + 1}{3 (\frac{1}{x + 1} (\frac{d}{d x} [x] + \frac{d}{d x} [1])) + \frac{d}{d x} [- 3 x]}$

Step 3.7.4

Differentiate using the Power Rule which states that

$\frac{d}{d x} [x^{n}]$ is

$n x^{n - 1}$ where

$n = 1$ .

$lim_{x \to 0} \frac{4 x + 1}{3 (\frac{1}{x + 1} (1 + \frac{d}{d x} [1])) + \frac{d}{d x} [- 3 x]}$

Step 3.7.5

Since

$1$ is constant with respect to

$x$ , the derivative of

$1$ with respect to

$x$ is

$0$ .

$lim_{x \to 0} \frac{4 x + 1}{3 (\frac{1}{x + 1} (1 + 0)) + \frac{d}{d x} [- 3 x]}$

Step 3.7.6

Add

$1$ and

$0$ .

$lim_{x \to 0} \frac{4 x + 1}{3 (\frac{1}{x + 1} \cdot 1) + \frac{d}{d x} [- 3 x]}$

Step 3.7.7

Multiply

$\frac{1}{x + 1}$ by

$1$ .

$lim_{x \to 0} \frac{4 x + 1}{3 \frac{1}{x + 1} + \frac{d}{d x} [- 3 x]}$

Step 3.7.8

Combine

$3$ and

$\frac{1}{x + 1}$ .

$lim_{x \to 0} \frac{4 x + 1}{\frac{3}{x + 1} + \frac{d}{d x} [- 3 x]}$

Step 3.8

Evaluate

$\frac{d}{d x} [- 3 x]$ .

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Step 3.8.1

Since

$- 3$ is constant with respect to

$x$ , the derivative of

$- 3 x$ with respect to

$x$ is

$- 3 \frac{d}{d x} [x]$ .

$lim_{x \to 0} \frac{4 x + 1}{\frac{3}{x + 1} - 3 \frac{d}{d x} [x]}$

Step 3.8.2

Differentiate using the Power Rule which states that

$\frac{d}{d x} [x^{n}]$ is

$n x^{n - 1}$ where

$n = 1$ .

$lim_{x \to 0} \frac{4 x + 1}{\frac{3}{x + 1} - 3 \cdot 1}$

Step 3.8.3

Multiply

$- 3$ by

$1$ .

$lim_{x \to 0} \frac{4 x + 1}{\frac{3}{x + 1} - 3}$

Step 3.9

Simplify.

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Step 3.9.1

Combine terms.

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Step 3.9.1.1

To write

$- 3$ as a fraction with a common denominator, multiply by

$\frac{x + 1}{x + 1}$ .

$lim_{x \to 0} \frac{4 x + 1}{\frac{3}{x + 1} - 3 \cdot \frac{x + 1}{x + 1}}$

Step 3.9.1.2

Combine

$- 3$ and

$\frac{x + 1}{x + 1}$ .

$lim_{x \to 0} \frac{4 x + 1}{\frac{3}{x + 1} + \frac{- 3 (x + 1)}{x + 1}}$

Step 3.9.1.3

Combine the numerators over the common denominator.

$lim_{x \to 0} \frac{4 x + 1}{\frac{3 - 3 (x + 1)}{x + 1}}$

Step 3.9.2

Simplify the numerator.

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Step 3.9.2.1

Factor

$3$ out of

$3 - 3 (x + 1)$ .

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Step 3.9.2.1.1

Factor

$3$ out of

$3$ .

$lim_{x \to 0} \frac{4 x + 1}{\frac{3 (1) - 3 (x + 1)}{x + 1}}$

Step 3.9.2.1.2

Factor

$3$ out of

$- 3 (x + 1)$ .

$lim_{x \to 0} \frac{4 x + 1}{\frac{3 (1) + 3 (- (x + 1))}{x + 1}}$

Step 3.9.2.1.3

Factor

$3$ out of

$3 (1) + 3 (- (x + 1))$ .

$lim_{x \to 0} \frac{4 x + 1}{\frac{3 (1 - (x + 1))}{x + 1}}$

Step 3.9.2.2

Apply the distributive property.

$lim_{x \to 0} \frac{4 x + 1}{\frac{3 (1 - x - 1 \cdot 1)}{x + 1}}$

Step 3.9.2.3

Multiply

$- 1$ by

$1$ .

$lim_{x \to 0} \frac{4 x + 1}{\frac{3 (1 - x - 1)}{x + 1}}$

Step 3.9.2.4

Subtract

$1$ from

$1$ .

$lim_{x \to 0} \frac{4 x + 1}{\frac{3 (- x + 0)}{x + 1}}$

Step 3.9.2.5

Add

$- x$ and

$0$ .

$lim_{x \to 0} \frac{4 x + 1}{\frac{3 \cdot - 1 x}{x + 1}}$

Step 3.9.2.6

Combine exponents.

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Step 3.9.2.6.1

Factor out negative.

$lim_{x \to 0} \frac{4 x + 1}{\frac{- (3 x)}{x + 1}}$

Step 3.9.2.6.2

Multiply

$3$ by

$- 1$ .

$lim_{x \to 0} \frac{4 x + 1}{\frac{- 3 x}{x + 1}}$

Step 3.9.3

Move the negative in front of the fraction.

$lim_{x \to 0} \frac{4 x + 1}{- \frac{3 x}{x + 1}}$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$lim_{x \to 0} (4 x + 1) (- \frac{x + 1}{3 x})$

Step 5

Consider the left sided limit.

$lim_{x \to 0^{-}} (4 x + 1) (- \frac{x + 1}{3 x})$

Step 6

As the

$x$ values approach

$0$ from the left, the function values increase without bound.

$\infty$

Step 7

Consider the right sided limit.

$lim_{x \to 0^{+}} (4 x + 1) (- \frac{x + 1}{3 x})$

Step 8

As the

$x$ values approach

$0$ from the right, the function values decrease without bound.

$- \infty$

Step 9

Since the left sided and right sided limits are not equal, the limit does not exist.

$Does not exist$