Calculus Examples

$lim_{x \to - 2} \frac{3 e^{- 2 - x} - 3}{2 x^{2} - 8}$

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.

$\frac{lim_{x \to - 2} 3 e^{- 2 - x} - 3}{lim_{x \to - 2} 2 x^{2} - 8}$

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 $- 2$ .

$\frac{lim_{x \to - 2} 3 e^{- 2 - x} - lim_{x \to - 2} 3}{lim_{x \to - 2} 2 x^{2} - 8}$

Step 1.2.2

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

$\frac{3 lim_{x \to - 2} e^{- 2 - x} - lim_{x \to - 2} 3}{lim_{x \to - 2} 2 x^{2} - 8}$

Step 1.2.3

Move the limit into the exponent.

$\frac{3 e^{lim_{x \to - 2} - 2 - x} - lim_{x \to - 2} 3}{lim_{x \to - 2} 2 x^{2} - 8}$

Step 1.2.4

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 2$ .

$\frac{3 e^{- lim_{x \to - 2} 2 - lim_{x \to - 2} x} - lim_{x \to - 2} 3}{lim_{x \to - 2} 2 x^{2} - 8}$

Step 1.2.5

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

$\frac{3 e^{- 1 \cdot 2 - lim_{x \to - 2} x} - lim_{x \to - 2} 3}{lim_{x \to - 2} 2 x^{2} - 8}$

Step 1.2.6

Evaluate the limit of $3$ which is constant as $x$ approaches $- 2$ .

$\frac{3 e^{- 1 \cdot 2 - lim_{x \to - 2} x} - 1 \cdot 3}{lim_{x \to - 2} 2 x^{2} - 8}$

Step 1.2.7

Simplify terms.

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Step 1.2.7.1

Evaluate the limit of $x$ by plugging in $- 2$ for $x$ .

$\frac{3 e^{- 1 \cdot 2 + 2} - 1 \cdot 3}{lim_{x \to - 2} 2 x^{2} - 8}$

Step 1.2.7.2

Simplify the answer.

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Step 1.2.7.2.1

Simplify each term.

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Step 1.2.7.2.1.1

Multiply $- 1$ by $2$ .

$\frac{3 e^{- 2 + 2} - 1 \cdot 3}{lim_{x \to - 2} 2 x^{2} - 8}$

Step 1.2.7.2.1.2

Add $- 2$ and $2$ .

$\frac{3 e^{0} - 1 \cdot 3}{lim_{x \to - 2} 2 x^{2} - 8}$

Step 1.2.7.2.1.3

Anything raised to $0$ is $1$ .

$\frac{3 \cdot 1 - 1 \cdot 3}{lim_{x \to - 2} 2 x^{2} - 8}$

Step 1.2.7.2.1.4

Multiply $3$ by $1$ .

$\frac{3 - 1 \cdot 3}{lim_{x \to - 2} 2 x^{2} - 8}$

Step 1.2.7.2.1.5

Multiply $- 1$ by $3$ .

$\frac{3 - 3}{lim_{x \to - 2} 2 x^{2} - 8}$

Step 1.2.7.2.2

Subtract $3$ from $3$ .

$\frac{0}{lim_{x \to - 2} 2 x^{2} - 8}$

Step 1.3

Evaluate the limit of the denominator.

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Step 1.3.1

Evaluate the limit.

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Step 1.3.1.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 2$ .

$\frac{0}{lim_{x \to - 2} 2 x^{2} - lim_{x \to - 2} 8}$

Step 1.3.1.2

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

$\frac{0}{2 lim_{x \to - 2} x^{2} - lim_{x \to - 2} 8}$

Step 1.3.1.3

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

$\frac{0}{2 {(lim_{x \to - 2} x)}^{2} - lim_{x \to - 2} 8}$

Step 1.3.1.4

Evaluate the limit of $8$ which is constant as $x$ approaches $- 2$ .

$\frac{0}{2 {(lim_{x \to - 2} x)}^{2} - 1 \cdot 8}$

Step 1.3.2

Evaluate the limit of $x$ by plugging in $- 2$ for $x$ .

$\frac{0}{2 {(- 2)}^{2} - 1 \cdot 8}$

Step 1.3.3

Simplify the answer.

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Step 1.3.3.1

Simplify each term.

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Step 1.3.3.1.1

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

$\frac{0}{2 \cdot 4 - 1 \cdot 8}$

Step 1.3.3.1.2

Multiply $2$ by $4$ .

$\frac{0}{8 - 1 \cdot 8}$

Step 1.3.3.1.3

Multiply $- 1$ by $8$ .

$\frac{0}{8 - 8}$

Step 1.3.3.2

Subtract $8$ from $8$ .

$\frac{0}{0}$

Step 1.3.3.3

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

Undefined

$\frac{0}{0}$

Step 1.3.4

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.

$lim_{x \to - 2} \frac{3 e^{- 2 - x} - 3}{2 x^{2} - 8} = lim_{x \to - 2} \frac{\frac{d}{d x} [3 e^{- 2 - x} - 3]}{\frac{d}{d x} [2 x^{2} - 8]}$

Step 3

Find the derivative of the numerator and denominator.

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Step 3.1

Differentiate the numerator and denominator.

$lim_{x \to - 2} \frac{\frac{d}{d x} [3 e^{- 2 - x} - 3]}{\frac{d}{d x} [2 x^{2} - 8]}$

Step 3.2

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

$lim_{x \to - 2} \frac{\frac{d}{d x} [3 e^{- 2 - x}] + \frac{d}{d x} [- 3]}{\frac{d}{d x} [2 x^{2} - 8]}$

Step 3.3

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

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Step 3.3.1

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

$lim_{x \to - 2} \frac{3 \frac{d}{d x} [e^{- 2 - x}] + \frac{d}{d x} [- 3]}{\frac{d}{d x} [2 x^{2} - 8]}$

Step 3.3.2

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

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Step 3.3.2.1

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

$lim_{x \to - 2} \frac{3 (\frac{d}{d u} [e^{u}] \frac{d}{d x} [- 2 - x]) + \frac{d}{d x} [- 3]}{\frac{d}{d x} [2 x^{2} - 8]}$

Step 3.3.2.2

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

$lim_{x \to - 2} \frac{3 (e^{u} \frac{d}{d x} [- 2 - x]) + \frac{d}{d x} [- 3]}{\frac{d}{d x} [2 x^{2} - 8]}$

Step 3.3.2.3

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

$lim_{x \to - 2} \frac{3 (e^{- 2 - x} \frac{d}{d x} [- 2 - x]) + \frac{d}{d x} [- 3]}{\frac{d}{d x} [2 x^{2} - 8]}$

Step 3.3.3

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

$lim_{x \to - 2} \frac{3 (e^{- 2 - x} (\frac{d}{d x} [- 2] + \frac{d}{d x} [- x])) + \frac{d}{d x} [- 3]}{\frac{d}{d x} [2 x^{2} - 8]}$

Step 3.3.4

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

$lim_{x \to - 2} \frac{3 (e^{- 2 - x} (0 + \frac{d}{d x} [- x])) + \frac{d}{d x} [- 3]}{\frac{d}{d x} [2 x^{2} - 8]}$

Step 3.3.5

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

$lim_{x \to - 2} \frac{3 (e^{- 2 - x} (0 - \frac{d}{d x} [x])) + \frac{d}{d x} [- 3]}{\frac{d}{d x} [2 x^{2} - 8]}$

Step 3.3.6

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

$lim_{x \to - 2} \frac{3 (e^{- 2 - x} (0 - 1 \cdot 1)) + \frac{d}{d x} [- 3]}{\frac{d}{d x} [2 x^{2} - 8]}$

Step 3.3.7

Multiply $- 1$ by $1$ .

$lim_{x \to - 2} \frac{3 (e^{- 2 - x} (0 - 1)) + \frac{d}{d x} [- 3]}{\frac{d}{d x} [2 x^{2} - 8]}$

Step 3.3.8

Subtract $1$ from $0$ .

$lim_{x \to - 2} \frac{3 (e^{- 2 - x} \cdot - 1) + \frac{d}{d x} [- 3]}{\frac{d}{d x} [2 x^{2} - 8]}$

Step 3.3.9

Move $- 1$ to the left of $e^{- 2 - x}$ .

$lim_{x \to - 2} \frac{3 (- 1 e^{- 2 - x}) + \frac{d}{d x} [- 3]}{\frac{d}{d x} [2 x^{2} - 8]}$

Step 3.3.10

Rewrite $- 1 e^{- 2 - x}$ as $- e^{- 2 - x}$ .

$lim_{x \to - 2} \frac{3 (- e^{- 2 - x}) + \frac{d}{d x} [- 3]}{\frac{d}{d x} [2 x^{2} - 8]}$

Step 3.3.11

Multiply $- 1$ by $3$ .

$lim_{x \to - 2} \frac{- 3 e^{- 2 - x} + \frac{d}{d x} [- 3]}{\frac{d}{d x} [2 x^{2} - 8]}$

Step 3.4

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

$lim_{x \to - 2} \frac{- 3 e^{- 2 - x} + 0}{\frac{d}{d x} [2 x^{2} - 8]}$

Step 3.5

Add $- 3 e^{- 2 - x}$ and $0$ .

$lim_{x \to - 2} \frac{- 3 e^{- 2 - x}}{\frac{d}{d x} [2 x^{2} - 8]}$

Step 3.6

By the Sum Rule, the derivative of $2 x^{2} - 8$ with respect to $x$ is $\frac{d}{d x} [2 x^{2}] + \frac{d}{d x} [- 8]$ .

$lim_{x \to - 2} \frac{- 3 e^{- 2 - x}}{\frac{d}{d x} [2 x^{2}] + \frac{d}{d x} [- 8]}$

Step 3.7

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

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Step 3.7.1

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

$lim_{x \to - 2} \frac{- 3 e^{- 2 - x}}{2 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 8]}$

Step 3.7.2

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

$lim_{x \to - 2} \frac{- 3 e^{- 2 - x}}{2 (2 x) + \frac{d}{d x} [- 8]}$

Step 3.7.3

Multiply $2$ by $2$ .

$lim_{x \to - 2} \frac{- 3 e^{- 2 - x}}{4 x + \frac{d}{d x} [- 8]}$

Step 3.8

Since $- 8$ is constant with respect to $x$ , the derivative of $- 8$ with respect to $x$ is $0$ .

$lim_{x \to - 2} \frac{- 3 e^{- 2 - x}}{4 x + 0}$

Step 3.9

Add $4 x$ and $0$ .

$lim_{x \to - 2} \frac{- 3 e^{- 2 - x}}{4 x}$

Step 4

Move the term $\frac{- 3}{4}$ outside of the limit because it is constant with respect to $x$ .

$\frac{- 3}{4} lim_{x \to - 2} \frac{e^{- 2 - x}}{x}$

Step 5

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $- 2$ .

$\frac{- 3}{4} \cdot \frac{lim_{x \to - 2} e^{- 2 - x}}{lim_{x \to - 2} x}$

Step 6

Move the limit into the exponent.

$\frac{- 3}{4} \cdot \frac{e^{lim_{x \to - 2} - 2 - x}}{lim_{x \to - 2} x}$

Step 7

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 2$ .

$\frac{- 3}{4} \cdot \frac{e^{- lim_{x \to - 2} 2 - lim_{x \to - 2} x}}{lim_{x \to - 2} x}$

Step 8

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

$\frac{- 3}{4} \cdot \frac{e^{- 1 \cdot 2 - lim_{x \to - 2} x}}{lim_{x \to - 2} x}$

Step 9

Evaluate the limits by plugging in $- 2$ for all occurrences of $x$ .

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Step 9.1

Evaluate the limit of $x$ by plugging in $- 2$ for $x$ .

$\frac{- 3}{4} \cdot \frac{e^{- 1 \cdot 2 + 2}}{lim_{x \to - 2} x}$

Step 9.2

Evaluate the limit of $x$ by plugging in $- 2$ for $x$ .

$\frac{- 3}{4} \cdot \frac{e^{- 1 \cdot 2 + 2}}{- 2}$

Step 10

Simplify the answer.

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Step 10.1

Combine.

$\frac{- 3 e^{- 1 \cdot 2 + 2}}{4 \cdot - 2}$

Step 10.2

Simplify the numerator.

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Step 10.2.1

Multiply $- 1$ by $2$ .

$\frac{- 3 e^{- 2 + 2}}{4 \cdot - 2}$

Step 10.2.2

Add $- 2$ and $2$ .

$\frac{- 3 e^{0}}{4 \cdot - 2}$

Step 10.2.3

Anything raised to $0$ is $1$ .

$\frac{- 3 \cdot 1}{4 \cdot - 2}$

Step 10.3

Multiply $4$ by $- 2$ .

$\frac{- 3 \cdot 1}{- 8}$

Step 10.4

Multiply $- 3$ by $1$ .

$\frac{- 3}{- 8}$

Step 10.5

Dividing two negative values results in a positive value.

$\frac{3}{8}$