Find the slant asymptote and the vertical asymptote. (If an answer does not exist, enter DNE, )

slant asymptote
vertical asymptote

Question

Find the slant asymptote and the vertical asymptote. (If an answer does not exist, enter DNE, ) $$ (x)=\frac{x^{2}}{x-7} $$ slant asymptote vertical asymptote

UpStudy AI · Accepted Answer

**Step 1: Identify the vertical asymptote**

The function is given by
$$
f(x)=\frac{x^2}{x-7}.
$$
A vertical asymptote occurs when the denominator is zero (and the numerator is nonzero). Set the denominator equal to zero:
$$
x-7=0.
$$
Solving for $$x$$, we get:
$$
x=7.
$$
Thus, the vertical asymptote is at $$x=7$$.

**Step 2: Find the slant asymptote**

Since the degree of the numerator (which is $$2$$) is exactly one more than the degree of the denominator (which is $$1$$), there is a slant asymptote. To find it, perform polynomial long division by dividing $$x^2$$ by $$x-7$$.

1. Divide $$x^2$$ by $$x$$ to get $$x$$.

2. Multiply $$x$$ by $$x-7$$ to get:
   $$
   x(x-7)=x^2-7x.
   $$

3. Subtract the result from $$x^2$$:
   $$
   x^2 - (x^2-7x)=7x.
   $$

4. Divide $$7x$$ by $$x$$ to get $$7$$.

5. Multiply $$7$$ by $$x-7$$ to get:
   $$
   7(x-7)=7x-49.
   $$

6. Subtract:
   $$
   7x - (7x-49)=49.
   $$

Thus, the division yields:
$$
\frac{x^2}{x-7} = x+7+\frac{49}{x-7}.
$$
As $$x$$ approaches $$\pm\infty$$, the term $$\frac{49}{x-7}$$ tends to $$0$$. Therefore, the slant asymptote is:
$$
y=x+7.
$$

**Final Answers:**

- **Slant asymptote:** $$y=x+7$$
- **Vertical asymptote:** $$x=7$$
Slant asymptote: $$ y = x + 7 $$ Vertical asymptote: $$ x = 7 $$

Find the slant asymptote and the vertical asymptote. (If an answer does not exist, enter DNE, )

slant asymptote
vertical asymptote

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Find the slant asymptote and the vertical asymptote. (If an answer does not exist, enter DNE, ) slant asymptote vertical asymptote

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Find the slant asymptote and the vertical asymptote. (If an answer does not exist, enter DNE, )

slant asymptote
vertical asymptote