Calculus Examples

$lim_{x \to 2} \frac{- 4 f (x)}{g (x)} + e^{x - 2} - 1$

Step 1

Evaluate the limit.

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Step 1.1

Cancel the common factor of $f (x)$ and $g (x)$ .

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Step 1.1.1

Cancel the common factor.

$lim_{x \to 2} \frac{- 4 f (x)}{g (x)} + e^{x - 2} - 1$

Step 1.1.2

Divide $- 4$ by $1$ .

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

Step 1.2

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

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

Step 1.3

Evaluate the limit of $4$ which is constant as $x$ approaches $2$ .

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

Step 1.4

Move the limit into the exponent.

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

Step 1.5

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

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

Step 1.6

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

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

Step 1.7

Evaluate the limit of $1$ which is constant as $x$ approaches $2$ .

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

Step 2

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

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

Step 3

Simplify the answer.

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Step 3.1

Simplify each term.

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Step 3.1.1

Multiply $- 1$ by $2$ .

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

Step 3.1.2

Subtract $2$ from $2$ .

$- 4 + e^{0} - 1 \cdot 1$

Step 3.1.3

Anything raised to $0$ is $1$ .

$- 4 + 1 - 1 \cdot 1$

Step 3.1.4

Multiply $- 1$ by $1$ .

$- 4 + 1 - 1$

Step 3.2

Add $- 4$ and $1$ .

$- 3 - 1$

Step 3.3

Subtract $1$ from $- 3$ .

$- 4$