Trigonometry Examples

$g (x) = 1000 \cdot 2^{\frac{x}{w}} + 500$

Step 1

Find where the expression $1000 \cdot 2^{\frac{x}{w}} + 500$ is undefined.

$x = 0$

Step 2

The vertical asymptotes occur at areas of infinite discontinuity.

No Vertical Asymptotes

Step 3

Evaluate $lim_{x \to - \infty} 1000 \cdot 2^{\frac{x}{w}} + 500$ to find the horizontal asymptote.

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Step 3.1

Evaluate the limit.

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Step 3.1.1

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

$lim_{x \to - \infty} 1000 \cdot 2^{\frac{x}{w}} + lim_{x \to - \infty} 500$

Step 3.1.2

Move the term $1000$ outside of the limit because it is constant with respect to $x$ .

$1000 lim_{x \to - \infty} 2^{\frac{x}{w}} + lim_{x \to - \infty} 500$

Step 3.2

Since the exponent $\frac{x}{w}$ approaches $- \infty$ , the quantity $2^{\frac{x}{w}}$ approaches $0$ .

$1000 \cdot 0 + lim_{x \to - \infty} 500$

Step 3.3

Evaluate the limit.

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Step 3.3.1

Evaluate the limit of $500$ which is constant as $x$ approaches $- \infty$ .

$1000 \cdot 0 + 500$

Step 3.3.2

Simplify the answer.

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Step 3.3.2.1

Multiply $1000$ by $0$ .

$0 + 500$

Step 3.3.2.2

Add $0$ and $500$ .

$500$

Step 4

List the horizontal asymptotes:

$y = 500$

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.

No Vertical Asymptotes

Horizontal Asymptotes: $y = 500$

No Oblique Asymptotes

Step 7