Calculus Examples

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

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.

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

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 $3$ .

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

Step 1.2.2

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

$\frac{{(lim_{x \to 3} x)}^{2} - lim_{x \to 3} 3 x}{lim_{x \to 3} 3 e^{2 x - 6} - x}$

Step 1.2.3

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

$\frac{{(lim_{x \to 3} x)}^{2} - 3 lim_{x \to 3} x}{lim_{x \to 3} 3 e^{2 x - 6} - x}$

Step 1.2.4

Evaluate the limits by plugging in $3$ for all occurrences of $x$ .

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Step 1.2.4.1

Evaluate the limit of $x$ by plugging in $3$ for $x$ .

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

Step 1.2.4.2

Evaluate the limit of $x$ by plugging in $3$ for $x$ .

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

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

Raise $3$ to the power of $2$ .

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

Step 1.2.5.1.2

Multiply $- 3$ by $3$ .

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

Step 1.2.5.2

Subtract $9$ from $9$ .

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

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 $3$ .

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

Step 1.3.2

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

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

Step 1.3.3

Move the limit into the exponent.

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

Step 1.3.4

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $3$ .

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

Step 1.3.5

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

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

Step 1.3.6

Evaluate the limit of $6$ which is constant as $x$ approaches $3$ .

$\frac{0}{3 e^{2 lim_{x \to 3} x - 1 \cdot 6} - lim_{x \to 3} x}$

Step 1.3.7

Evaluate the limits by plugging in $3$ for all occurrences of $x$ .

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Step 1.3.7.1

Evaluate the limit of $x$ by plugging in $3$ for $x$ .

$\frac{0}{3 e^{2 \cdot 3 - 1 \cdot 6} - lim_{x \to 3} x}$

Step 1.3.7.2

Evaluate the limit of $x$ by plugging in $3$ for $x$ .

$\frac{0}{3 e^{2 \cdot 3 - 1 \cdot 6} - 3}$

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

Simplify each term.

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Step 1.3.8.1.1.1

Multiply $2$ by $3$ .

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

Step 1.3.8.1.1.2

Multiply $- 1$ by $6$ .

$\frac{0}{3 e^{6 - 6} - 3}$

Step 1.3.8.1.2

Subtract $6$ from $6$ .

$\frac{0}{3 e^{0} - 3}$

Step 1.3.8.1.3

Anything raised to $0$ is $1$ .

$\frac{0}{3 \cdot 1 - 3}$

Step 1.3.8.1.4

Multiply $3$ by $1$ .

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

Step 1.3.8.2

Subtract $3$ from $3$ .

$\frac{0}{0}$

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.

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

Step 3

Find the derivative of the numerator and denominator.

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Step 3.1

Differentiate the numerator and denominator.

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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Step 3.4.1

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

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

Step 3.4.2

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

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

Step 3.4.3

Multiply $- 3$ by $1$ .

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

Step 3.5

By the Sum Rule, the derivative of $3 e^{2 x - 6} - x$ with respect to $x$ is $\frac{d}{d x} [3 e^{2 x - 6}] + \frac{d}{d x} [- x]$ .

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

Step 3.6

Evaluate $\frac{d}{d x} [3 e^{2 x - 6}]$ .

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Step 3.6.1

Since $3$ is constant with respect to $x$ , the derivative of $3 e^{2 x - 6}$ with respect to $x$ is $3 \frac{d}{d x} [e^{2 x - 6}]$ .

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

Step 3.6.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = 2 x - 6$ .

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Step 3.6.2.1

To apply the Chain Rule, set $u$ as $2 x - 6$ .

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

Step 3.6.2.2

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

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

Step 3.6.2.3

Replace all occurrences of $u$ with $2 x - 6$ .

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

Step 3.6.3

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

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

Step 3.6.4

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

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

Step 3.6.5

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

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

Step 3.6.6

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6$ with respect to $x$ is $0$ .

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

Step 3.6.7

Multiply $2$ by $1$ .

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

Step 3.6.8

Add $2$ and $0$ .

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

Step 3.6.9

Move $2$ to the left of $e^{2 x - 6}$ .

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

Step 3.6.10

Multiply $2$ by $3$ .

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

Step 3.7

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

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Step 3.7.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x$ with respect to $x$ is $- \frac{d}{d x} [x]$ .

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

Step 3.7.2

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

$lim_{x \to 3} \frac{2 x - 3}{6 e^{2 x - 6} - 1 \cdot 1}$

Step 3.7.3

Multiply $- 1$ by $1$ .

$lim_{x \to 3} \frac{2 x - 3}{6 e^{2 x - 6} - 1}$

Step 4

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $3$ .

$\frac{lim_{x \to 3} 2 x - 3}{lim_{x \to 3} 6 e^{2 x - 6} - 1}$

Step 5

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $3$ .

$\frac{lim_{x \to 3} 2 x - lim_{x \to 3} 3}{lim_{x \to 3} 6 e^{2 x - 6} - 1}$

Step 6

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

$\frac{2 lim_{x \to 3} x - lim_{x \to 3} 3}{lim_{x \to 3} 6 e^{2 x - 6} - 1}$

Step 7

Evaluate the limit of $3$ which is constant as $x$ approaches $3$ .

$\frac{2 lim_{x \to 3} x - 1 \cdot 3}{lim_{x \to 3} 6 e^{2 x - 6} - 1}$

Step 8

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $3$ .

$\frac{2 lim_{x \to 3} x - 1 \cdot 3}{lim_{x \to 3} 6 e^{2 x - 6} - lim_{x \to 3} 1}$

Step 9

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

$\frac{2 lim_{x \to 3} x - 1 \cdot 3}{6 lim_{x \to 3} e^{2 x - 6} - lim_{x \to 3} 1}$

Step 10

Move the limit into the exponent.

$\frac{2 lim_{x \to 3} x - 1 \cdot 3}{6 e^{lim_{x \to 3} 2 x - 6} - lim_{x \to 3} 1}$

Step 11

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $3$ .

$\frac{2 lim_{x \to 3} x - 1 \cdot 3}{6 e^{lim_{x \to 3} 2 x - lim_{x \to 3} 6} - lim_{x \to 3} 1}$

Step 12

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

$\frac{2 lim_{x \to 3} x - 1 \cdot 3}{6 e^{2 lim_{x \to 3} x - lim_{x \to 3} 6} - lim_{x \to 3} 1}$

Step 13

Evaluate the limit of $6$ which is constant as $x$ approaches $3$ .

$\frac{2 lim_{x \to 3} x - 1 \cdot 3}{6 e^{2 lim_{x \to 3} x - 1 \cdot 6} - lim_{x \to 3} 1}$

Step 14

Evaluate the limit of $1$ which is constant as $x$ approaches $3$ .

$\frac{2 lim_{x \to 3} x - 1 \cdot 3}{6 e^{2 lim_{x \to 3} x - 1 \cdot 6} - 1 \cdot 1}$

Step 15

Evaluate the limits by plugging in $3$ for all occurrences of $x$ .

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Step 15.1

Evaluate the limit of $x$ by plugging in $3$ for $x$ .

$\frac{2 \cdot 3 - 1 \cdot 3}{6 e^{2 lim_{x \to 3} x - 1 \cdot 6} - 1 \cdot 1}$

Step 15.2

Evaluate the limit of $x$ by plugging in $3$ for $x$ .

$\frac{2 \cdot 3 - 1 \cdot 3}{6 e^{2 \cdot 3 - 1 \cdot 6} - 1 \cdot 1}$

Step 16

Simplify the answer.

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Step 16.1

Simplify the numerator.

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Step 16.1.1

Multiply $2$ by $3$ .

$\frac{6 - 1 \cdot 3}{6 e^{2 \cdot 3 - 1 \cdot 6} - 1 \cdot 1}$

Step 16.1.2

Multiply $- 1$ by $3$ .

$\frac{6 - 3}{6 e^{2 \cdot 3 - 1 \cdot 6} - 1 \cdot 1}$

Step 16.1.3

Subtract $3$ from $6$ .

$\frac{3}{6 e^{2 \cdot 3 - 1 \cdot 6} - 1 \cdot 1}$

Step 16.2

Simplify the denominator.

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Step 16.2.1

Simplify each term.

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Step 16.2.1.1

Multiply $2$ by $3$ .

$\frac{3}{6 e^{6 - 1 \cdot 6} - 1 \cdot 1}$

Step 16.2.1.2

Multiply $- 1$ by $6$ .

$\frac{3}{6 e^{6 - 6} - 1 \cdot 1}$

Step 16.2.2

Subtract $6$ from $6$ .

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

Step 16.2.3

Anything raised to $0$ is $1$ .

$\frac{3}{6 \cdot 1 - 1 \cdot 1}$

Step 16.2.4

Multiply $6$ by $1$ .

$\frac{3}{6 - 1 \cdot 1}$

Step 16.2.5

Multiply $- 1$ by $1$ .

$\frac{3}{6 - 1}$

Step 16.2.6

Subtract $1$ from $6$ .

$\frac{3}{5}$