Trigonometry Examples

$3 x^{2 - 10 x - 8}$

Step 1

Find where the expression $3 x^{- 10 x - 6}$ is undefined.

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Step 2

The vertical asymptotes occur at areas of infinite discontinuity.

No Vertical Asymptotes

Step 3

Evaluate $lim_{x \to \infty} 3 x^{- 10 x - 6}$ to find the horizontal asymptote.

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Step 3.1

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

$3 lim_{x \to \infty} x^{- 10 x - 6}$

Step 3.2

Use the properties of logarithms to simplify the limit.

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Step 3.2.1

Rewrite $x^{- 10 x - 6}$ as $e^{\ln (x^{- 10 x - 6})}$ .

$3 lim_{x \to \infty} e^{\ln (x^{- 10 x - 6})}$

Step 3.2.2

Expand $\ln (x^{- 10 x - 6})$ by moving $- 10 x - 6$ outside the logarithm.

$3 lim_{x \to \infty} e^{(- 10 x - 6) \ln (x)}$

Step 3.3

Since the exponent $(- 10 x - 6) \ln (x)$ approaches $- \infty$ , the quantity $e^{(- 10 x - 6) \ln (x)}$ approaches $0$ .

$3 \cdot 0$

Step 3.4

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.

No Vertical Asymptotes

Horizontal Asymptotes: $y = 0$

No Oblique Asymptotes

Step 7