Pre-Algebra Examples

$x^{3 (2 x - 6 x^{2})}$

Step 1

Find where the expression $x^{6 x - 18 x^{2}}$ 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 $lim_{x \to \infty} x^{6 x - 18 x^{2}}$ to find the horizontal asymptote.

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Step 3.1

Use the properties of logarithms to simplify the limit.

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Step 3.1.1

Rewrite $x^{6 x - 18 x^{2}}$ as $e^{\ln (x^{6 x - 18 x^{2}})}$ .

$lim_{x \to \infty} e^{\ln (x^{6 x - 18 x^{2}})}$

Step 3.1.2

Expand $\ln (x^{6 x - 18 x^{2}})$ by moving $6 x - 18 x^{2}$ outside the logarithm.

$lim_{x \to \infty} e^{(6 x - 18 x^{2}) \ln (x)}$

Step 3.2

Since the exponent $(6 x - 18 x^{2}) \ln (x)$ approaches $- \infty$ , the quantity $e^{(6 x - 18 x^{2}) \ln (x)}$ approaches $0$ .

$0$

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