Calculus Examples

$\frac{lim_{x \to 0} e^{x^{2} + 3 x} - 1}{e^{x^{2} + 4 x} - 1}$

Step 1

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

$\frac{lim_{x \to 0} e^{x^{2} + 3 x} - lim_{x \to 0} 1}{e^{x^{2} + 4 x} - 1}$

Step 2

Move the limit into the exponent.

$\frac{e^{lim_{x \to 0} x^{2} + 3 x} - lim_{x \to 0} 1}{e^{x^{2} + 4 x} - 1}$

Step 3

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

$\frac{e^{lim_{x \to 0} x^{2} + lim_{x \to 0} 3 x} - lim_{x \to 0} 1}{e^{x^{2} + 4 x} - 1}$

Step 4

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

$\frac{e^{{(lim_{x \to 0} x)}^{2} + lim_{x \to 0} 3 x} - lim_{x \to 0} 1}{e^{x^{2} + 4 x} - 1}$

Step 5

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

$\frac{e^{{(lim_{x \to 0} x)}^{2} + 3 lim_{x \to 0} x} - lim_{x \to 0} 1}{e^{x^{2} + 4 x} - 1}$

Step 6

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

$\frac{e^{{(lim_{x \to 0} x)}^{2} + 3 lim_{x \to 0} x} - 1 \cdot 1}{e^{x^{2} + 4 x} - 1}$

Step 7

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

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Step 7.1

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

$\frac{e^{0^{2} + 3 lim_{x \to 0} x} - 1 \cdot 1}{e^{x^{2} + 4 x} - 1}$

Step 7.2

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

$\frac{e^{0^{2} + 3 \cdot 0} - 1 \cdot 1}{e^{x^{2} + 4 x} - 1}$

Step 8

Simplify the answer.

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Step 8.1

Simplify the numerator.

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Step 8.1.1

Simplify each term.

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Step 8.1.1.1

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

$\frac{e^{0 + 3 \cdot 0} - 1 \cdot 1}{e^{x^{2} + 4 x} - 1}$

Step 8.1.1.2

Multiply $3$ by $0$ .

$\frac{e^{0 + 0} - 1 \cdot 1}{e^{x^{2} + 4 x} - 1}$

Step 8.1.2

Add $0$ and $0$ .

$\frac{e^{0} - 1 \cdot 1}{e^{x^{2} + 4 x} - 1}$

Step 8.1.3

Anything raised to $0$ is $1$ .

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

Step 8.1.4

Multiply $- 1$ by $1$ .

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

Step 8.1.5

Subtract $1$ from $1$ .

$\frac{0}{e^{x^{2} + 4 x} - 1}$

Step 8.2

Divide $0$ by $e^{x^{2} + 4 x} - 1$ .

$0$