Trigonometry Examples

$h (x) e^{x + 1} + 3$

Step 1

Simplify.

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Step 1.1

Subtract $3$ from both sides of the equation.

$y x e^{x + 1} = - 3$

Step 1.2

Divide each term in $y x e^{x + 1} = - 3$ by $x e^{x + 1}$ and simplify.

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Step 1.2.1

Divide each term in $y x e^{x + 1} = - 3$ by $x e^{x + 1}$ .

$\frac{y x e^{x + 1}}{x e^{x + 1}} = \frac{- 3}{x e^{x + 1}}$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $x$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{y x e^{x + 1}}{x e^{x + 1}} = \frac{- 3}{x e^{x + 1}}$

Step 1.2.2.1.2

Rewrite the expression.

$\frac{y e^{x + 1}}{e^{x + 1}} = \frac{- 3}{x e^{x + 1}}$

Step 1.2.2.2

Cancel the common factor of $e^{x + 1}$ .

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Step 1.2.2.2.1

Cancel the common factor.

$\frac{y e^{x + 1}}{e^{x + 1}} = \frac{- 3}{x e^{x + 1}}$

Step 1.2.2.2.2

Divide $y$ by $1$ .

$y = \frac{- 3}{x e^{x + 1}}$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Move the negative in front of the fraction.

$y = - \frac{3}{x e^{x + 1}}$

Step 2

Find where the expression $- \frac{3}{x e^{x + 1}}$ is undefined.

$x = 0$

Step 3

Evaluate $lim_{x \to \infty} - \frac{3}{x e^{x + 1}}$ to find the horizontal asymptote.

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Step 3.1

Evaluate the limit.

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Step 3.1.1

Move the term $- 1$ outside of the limit because it is constant with respect to $x$ .

$- lim_{x \to \infty} \frac{3}{x e^{x + 1}}$

Step 3.1.2

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

$- 3 lim_{x \to \infty} \frac{1}{x e^{x + 1}}$

Step 3.2

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{x e^{x + 1}}$ approaches $0$ .

$- 3 \cdot 0$

Step 3.3

Multiply $- 3$ by $0$ .

$0$

Step 4

List the horizontal asymptotes:

$y = 0$

Step 5

There is no oblique asymptote because the degree of the numerator is less than or equal to the degree of the denominator.

No Oblique Asymptotes

Step 6

This is the set of all asymptotes.

Vertical Asymptotes: $x = 0$

Horizontal Asymptotes: $y = 0$

No Oblique Asymptotes

Step 7