Calculus Examples

$lim_{x \to 0} \frac{6 x e^{- 3 x + 9}}{6 x^{2} + x}$

Step 1

Move the term $6$ outside of the limit because it is constant with respect to $x$ .

$6 lim_{x \to 0} \frac{x e^{- 3 x + 9}}{6 x^{2} + x}$

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.

$6 \frac{lim_{x \to 0} x e^{- 3 x + 9}}{lim_{x \to 0} 6 x^{2} + x}$

Step 2.1.2

Evaluate the limit of the numerator.

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Step 2.1.2.1

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

$6 \frac{lim_{x \to 0} x \cdot lim_{x \to 0} e^{- 3 x + 9}}{lim_{x \to 0} 6 x^{2} + x}$

Step 2.1.2.2

Move the limit into the exponent.

$6 \frac{lim_{x \to 0} x \cdot e^{lim_{x \to 0} - 3 x + 9}}{lim_{x \to 0} 6 x^{2} + x}$

Step 2.1.2.3

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

$6 \frac{lim_{x \to 0} x \cdot e^{- lim_{x \to 0} 3 x + lim_{x \to 0} 9}}{lim_{x \to 0} 6 x^{2} + x}$

Step 2.1.2.4

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

$6 \frac{lim_{x \to 0} x \cdot e^{- 3 lim_{x \to 0} x + lim_{x \to 0} 9}}{lim_{x \to 0} 6 x^{2} + x}$

Step 2.1.2.5

Evaluate the limit of $9$ which is constant as $x$ approaches $0$ .

$6 \frac{lim_{x \to 0} x \cdot e^{- 3 lim_{x \to 0} x + 9}}{lim_{x \to 0} 6 x^{2} + x}$

Step 2.1.2.6

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

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Step 2.1.2.6.1

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

$6 \frac{0 \cdot e^{- 3 lim_{x \to 0} x + 9}}{lim_{x \to 0} 6 x^{2} + x}$

Step 2.1.2.6.2

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

$6 \frac{0 \cdot e^{- 3 \cdot 0 + 9}}{lim_{x \to 0} 6 x^{2} + x}$

Step 2.1.2.7

Simplify the answer.

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Step 2.1.2.7.1

Multiply $- 3$ by $0$ .

$6 \frac{0 \cdot e^{0 + 9}}{lim_{x \to 0} 6 x^{2} + x}$

Step 2.1.2.7.2

Add $0$ and $9$ .

$6 \frac{0 \cdot e^{9}}{lim_{x \to 0} 6 x^{2} + x}$

Step 2.1.2.7.3

Multiply $0$ by $e^{9}$ .

$6 \frac{0}{lim_{x \to 0} 6 x^{2} + x}$

Step 2.1.3

Evaluate the limit of the denominator.

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Step 2.1.3.1

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

$6 \frac{0}{lim_{x \to 0} 6 x^{2} + lim_{x \to 0} x}$

Step 2.1.3.2

Move the term $6$ outside of the limit because it is constant with respect to $x$ .

$6 \frac{0}{6 lim_{x \to 0} x^{2} + lim_{x \to 0} x}$

Step 2.1.3.3

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

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

Step 2.1.3.4

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

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Step 2.1.3.4.1

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

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

Step 2.1.3.4.2

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

$6 \frac{0}{6 \cdot 0^{2} + 0}$

Step 2.1.3.5

Simplify the answer.

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Step 2.1.3.5.1

Simplify each term.

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Step 2.1.3.5.1.1

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

$6 \frac{0}{6 \cdot 0 + 0}$

Step 2.1.3.5.1.2

Multiply $6$ by $0$ .

$6 \frac{0}{0 + 0}$

Step 2.1.3.5.2

Add $0$ and $0$ .

$6 (\frac{0}{0})$

Step 2.1.3.5.3

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

Undefined

$6 (\frac{0}{0})$

Step 2.1.3.6

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

Undefined

$6 (\frac{0}{0})$

Step 2.1.4

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

Undefined

$6 (\frac{0}{0})$

Step 2.2

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.

$lim_{x \to 0} \frac{x e^{- 3 x + 9}}{6 x^{2} + x} = lim_{x \to 0} \frac{\frac{d}{d x} [x e^{- 3 x + 9}]}{\frac{d}{d x} [6 x^{2} + x]}$

Step 2.3

Find the derivative of the numerator and denominator.

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Step 2.3.1

Differentiate the numerator and denominator.

$6 lim_{x \to 0} \frac{\frac{d}{d x} [x e^{- 3 x + 9}]}{\frac{d}{d x} [6 x^{2} + x]}$

Step 2.3.2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = e^{- 3 x + 9}$ .

$6 lim_{x \to 0} \frac{x \frac{d}{d x} [e^{- 3 x + 9}] + e^{- 3 x + 9} \frac{d}{d x} [x]}{\frac{d}{d x} [6 x^{2} + x]}$

Step 2.3.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = - 3 x + 9$ .

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Step 2.3.3.1

To apply the Chain Rule, set $u$ as $- 3 x + 9$ .

$6 lim_{x \to 0} \frac{x \frac{d}{d u} [e^{u}] \frac{d}{d x} [- 3 x + 9] + e^{- 3 x + 9} \frac{d}{d x} [x]}{\frac{d}{d x} [6 x^{2} + x]}$

Step 2.3.3.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u} [a^{u}]$ is $a^{u} \ln (a)$ where $a$ = $e$ .

$6 lim_{x \to 0} \frac{x e^{u} \frac{d}{d x} [- 3 x + 9] + e^{- 3 x + 9} \frac{d}{d x} [x]}{\frac{d}{d x} [6 x^{2} + x]}$

Step 2.3.3.3

Replace all occurrences of $u$ with $- 3 x + 9$ .

$6 lim_{x \to 0} \frac{x e^{- 3 x + 9} \frac{d}{d x} [- 3 x + 9] + e^{- 3 x + 9} \frac{d}{d x} [x]}{\frac{d}{d x} [6 x^{2} + x]}$

Step 2.3.4

By the Sum Rule, the derivative of $- 3 x + 9$ with respect to $x$ is $\frac{d}{d x} [- 3 x] + \frac{d}{d x} [9]$ .

$6 lim_{x \to 0} \frac{x e^{- 3 x + 9} (\frac{d}{d x} [- 3 x] + \frac{d}{d x} [9]) + e^{- 3 x + 9} \frac{d}{d x} [x]}{\frac{d}{d x} [6 x^{2} + x]}$

Step 2.3.5

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

$6 lim_{x \to 0} \frac{x e^{- 3 x + 9} (- 3 \frac{d}{d x} [x] + \frac{d}{d x} [9]) + e^{- 3 x + 9} \frac{d}{d x} [x]}{\frac{d}{d x} [6 x^{2} + x]}$

Step 2.3.6

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

$6 lim_{x \to 0} \frac{x e^{- 3 x + 9} (- 3 \cdot 1 + \frac{d}{d x} [9]) + e^{- 3 x + 9} \frac{d}{d x} [x]}{\frac{d}{d x} [6 x^{2} + x]}$

Step 2.3.7

Multiply $- 3$ by $1$ .

$6 lim_{x \to 0} \frac{x e^{- 3 x + 9} (- 3 + \frac{d}{d x} [9]) + e^{- 3 x + 9} \frac{d}{d x} [x]}{\frac{d}{d x} [6 x^{2} + x]}$

Step 2.3.8

Since $9$ is constant with respect to $x$ , the derivative of $9$ with respect to $x$ is $0$ .

$6 lim_{x \to 0} \frac{x e^{- 3 x + 9} (- 3 + 0) + e^{- 3 x + 9} \frac{d}{d x} [x]}{\frac{d}{d x} [6 x^{2} + x]}$

Step 2.3.9

Add $- 3$ and $0$ .

$6 lim_{x \to 0} \frac{x e^{- 3 x + 9} \cdot - 3 + e^{- 3 x + 9} \frac{d}{d x} [x]}{\frac{d}{d x} [6 x^{2} + x]}$

Step 2.3.10

Move $- 3$ to the left of $e^{- 3 x + 9}$ .

$6 lim_{x \to 0} \frac{x \cdot - 3 \cdot e^{- 3 x + 9} + e^{- 3 x + 9} \frac{d}{d x} [x]}{\frac{d}{d x} [6 x^{2} + x]}$

Step 2.3.11

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

$6 lim_{x \to 0} \frac{x \cdot - 3 e^{- 3 x + 9} + e^{- 3 x + 9} \cdot 1}{\frac{d}{d x} [6 x^{2} + x]}$

Step 2.3.12

Multiply $e^{- 3 x + 9}$ by $1$ .

$6 lim_{x \to 0} \frac{x \cdot - 3 e^{- 3 x + 9} + e^{- 3 x + 9}}{\frac{d}{d x} [6 x^{2} + x]}$

Step 2.3.13

Simplify.

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Step 2.3.13.1

Reorder terms.

$6 lim_{x \to 0} \frac{- 3 e^{- 3 x + 9} x + e^{- 3 x + 9}}{\frac{d}{d x} [6 x^{2} + x]}$

Step 2.3.13.2

Reorder factors in $- 3 e^{- 3 x + 9} x + e^{- 3 x + 9}$ .

$6 lim_{x \to 0} \frac{- 3 x e^{- 3 x + 9} + e^{- 3 x + 9}}{\frac{d}{d x} [6 x^{2} + x]}$

Step 2.3.14

By the Sum Rule, the derivative of $6 x^{2} + x$ with respect to $x$ is $\frac{d}{d x} [6 x^{2}] + \frac{d}{d x} [x]$ .

$6 lim_{x \to 0} \frac{- 3 x e^{- 3 x + 9} + e^{- 3 x + 9}}{\frac{d}{d x} [6 x^{2}] + \frac{d}{d x} [x]}$

Step 2.3.15

Evaluate $\frac{d}{d x} [6 x^{2}]$ .

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Step 2.3.15.1

Since $6$ is constant with respect to $x$ , the derivative of $6 x^{2}$ with respect to $x$ is $6 \frac{d}{d x} [x^{2}]$ .

$6 lim_{x \to 0} \frac{- 3 x e^{- 3 x + 9} + e^{- 3 x + 9}}{6 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [x]}$

Step 2.3.15.2

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

$6 lim_{x \to 0} \frac{- 3 x e^{- 3 x + 9} + e^{- 3 x + 9}}{6 \cdot 2 x + \frac{d}{d x} [x]}$

Step 2.3.15.3

Multiply $2$ by $6$ .

$6 lim_{x \to 0} \frac{- 3 x e^{- 3 x + 9} + e^{- 3 x + 9}}{12 x + \frac{d}{d x} [x]}$

Step 2.3.16

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

$6 lim_{x \to 0} \frac{- 3 x e^{- 3 x + 9} + e^{- 3 x + 9}}{12 x + 1}$

Step 3

Evaluate the limit.

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Step 3.1

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

$6 \frac{lim_{x \to 0} - 3 x e^{- 3 x + 9} + e^{- 3 x + 9}}{lim_{x \to 0} 12 x + 1}$

Step 3.2

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

$6 \frac{- lim_{x \to 0} 3 x e^{- 3 x + 9} + lim_{x \to 0} e^{- 3 x + 9}}{lim_{x \to 0} 12 x + 1}$

Step 3.3

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

$6 \frac{- 3 lim_{x \to 0} x e^{- 3 x + 9} + lim_{x \to 0} e^{- 3 x + 9}}{lim_{x \to 0} 12 x + 1}$

Step 3.4

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

$6 \frac{- 3 lim_{x \to 0} x \cdot lim_{x \to 0} e^{- 3 x + 9} + lim_{x \to 0} e^{- 3 x + 9}}{lim_{x \to 0} 12 x + 1}$

Step 3.5

Move the limit into the exponent.

$6 \frac{- 3 lim_{x \to 0} x \cdot e^{lim_{x \to 0} - 3 x + 9} + lim_{x \to 0} e^{- 3 x + 9}}{lim_{x \to 0} 12 x + 1}$

Step 3.6

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

$6 \frac{- 3 lim_{x \to 0} x \cdot e^{- lim_{x \to 0} 3 x + lim_{x \to 0} 9} + lim_{x \to 0} e^{- 3 x + 9}}{lim_{x \to 0} 12 x + 1}$

Step 3.7

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

$6 \frac{- 3 lim_{x \to 0} x \cdot e^{- 3 lim_{x \to 0} x + lim_{x \to 0} 9} + lim_{x \to 0} e^{- 3 x + 9}}{lim_{x \to 0} 12 x + 1}$

Step 3.8

Evaluate the limit of $9$ which is constant as $x$ approaches $0$ .

$6 \frac{- 3 lim_{x \to 0} x \cdot e^{- 3 lim_{x \to 0} x + 9} + lim_{x \to 0} e^{- 3 x + 9}}{lim_{x \to 0} 12 x + 1}$

Step 3.9

Move the limit into the exponent.

$6 \frac{- 3 lim_{x \to 0} x \cdot e^{- 3 lim_{x \to 0} x + 9} + e^{lim_{x \to 0} - 3 x + 9}}{lim_{x \to 0} 12 x + 1}$

Step 3.10

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

$6 \frac{- 3 lim_{x \to 0} x \cdot e^{- 3 lim_{x \to 0} x + 9} + e^{- lim_{x \to 0} 3 x + lim_{x \to 0} 9}}{lim_{x \to 0} 12 x + 1}$

Step 3.11

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

$6 \frac{- 3 lim_{x \to 0} x \cdot e^{- 3 lim_{x \to 0} x + 9} + e^{- 3 lim_{x \to 0} x + lim_{x \to 0} 9}}{lim_{x \to 0} 12 x + 1}$

Step 3.12

Evaluate the limit of $9$ which is constant as $x$ approaches $0$ .

$6 \frac{- 3 lim_{x \to 0} x \cdot e^{- 3 lim_{x \to 0} x + 9} + e^{- 3 lim_{x \to 0} x + 9}}{lim_{x \to 0} 12 x + 1}$

Step 3.13

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

$6 \frac{- 3 lim_{x \to 0} x \cdot e^{- 3 lim_{x \to 0} x + 9} + e^{- 3 lim_{x \to 0} x + 9}}{lim_{x \to 0} 12 x + lim_{x \to 0} 1}$

Step 3.14

Move the term $12$ outside of the limit because it is constant with respect to $x$ .

$6 \frac{- 3 lim_{x \to 0} x \cdot e^{- 3 lim_{x \to 0} x + 9} + e^{- 3 lim_{x \to 0} x + 9}}{12 lim_{x \to 0} x + lim_{x \to 0} 1}$

Step 3.15

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

$6 \frac{- 3 lim_{x \to 0} x \cdot e^{- 3 lim_{x \to 0} x + 9} + e^{- 3 lim_{x \to 0} x + 9}}{12 lim_{x \to 0} x + 1}$

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 $x$ .

$6 \frac{- 3 \cdot 0 \cdot e^{- 3 lim_{x \to 0} x + 9} + e^{- 3 lim_{x \to 0} x + 9}}{12 lim_{x \to 0} x + 1}$

Step 4.2

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

$6 \frac{- 3 \cdot 0 \cdot e^{- 3 \cdot 0 + 9} + e^{- 3 lim_{x \to 0} x + 9}}{12 lim_{x \to 0} x + 1}$

Step 4.3

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

$6 \frac{- 3 \cdot 0 \cdot e^{- 3 \cdot 0 + 9} + e^{- 3 \cdot 0 + 9}}{12 lim_{x \to 0} x + 1}$

Step 4.4

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

$6 \frac{- 3 \cdot 0 \cdot e^{- 3 \cdot 0 + 9} + e^{- 3 \cdot 0 + 9}}{12 \cdot 0 + 1}$

Step 5

Simplify the answer.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Multiply $- 3$ by $0$ .

$6 \frac{0 \cdot e^{- 3 \cdot 0 + 9} + e^{- 3 \cdot 0 + 9}}{12 \cdot 0 + 1}$

Step 5.1.2

Multiply $- 3$ by $0$ .

$6 \frac{0 \cdot e^{0 + 9} + e^{- 3 \cdot 0 + 9}}{12 \cdot 0 + 1}$

Step 5.1.3

Add $0$ and $9$ .

$6 \frac{0 \cdot e^{9} + e^{- 3 \cdot 0 + 9}}{12 \cdot 0 + 1}$

Step 5.1.4

Multiply $0$ by $e^{9}$ .

$6 \frac{0 + e^{- 3 \cdot 0 + 9}}{12 \cdot 0 + 1}$

Step 5.1.5

Multiply $- 3$ by $0$ .

$6 \frac{0 + e^{0 + 9}}{12 \cdot 0 + 1}$

Step 5.1.6

Add $0$ and $9$ .

$6 \frac{0 + e^{9}}{12 \cdot 0 + 1}$

Step 5.1.7

Add $0$ and $e^{9}$ .

$6 \frac{e^{9}}{12 \cdot 0 + 1}$

Step 5.2

Simplify the denominator.

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Step 5.2.1

Multiply $12$ by $0$ .

$6 \frac{e^{9}}{0 + 1}$

Step 5.2.2

Add $0$ and $1$ .

$6 \frac{e^{9}}{1}$

Step 5.3

Divide $e^{9}$ by $1$ .

$6 e^{9}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$6 e^{9}$

Decimal Form:

$48618.50356545 \dots$