# How do you find vertical, horizontal and oblique asymptotes for #y = (x + 1)/(x - 1)#?

vertical asymptote x = 1

horizontal asymptote y = 1

When the denominator of a rational function tends to zero, vertical asymptotes occur. Let the denominator equal zero to find the equation.

solve: x - 1 = 0 → the asymptote is x = 1.

Divide each term by x on the denominator and numerator.

There are no oblique asymptotes in this case because oblique asymptotes arise when the degree of the numerator is greater than the degree of the denominator.

The function's graph is shown here: graph{(x+1)/(x-1) [-10, 10, -5, 5]}

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To find the vertical asymptotes of the function y = (x + 1)/(x - 1), we need to determine where the denominator becomes zero, because division by zero is undefined. Setting the denominator equal to zero, we find:

x - 1 = 0 x = 1

Therefore, x = 1 is a vertical asymptote.

To find horizontal asymptotes, we analyze the behavior of the function as x approaches positive or negative infinity. We use the concept of limits.

As x approaches positive infinity, the function becomes:

lim (x -> ∞) (x + 1) / (x - 1)

By dividing the leading terms, we get:

lim (x -> ∞) (x/x) = 1

Similarly, as x approaches negative infinity:

lim (x -> -∞) (x + 1) / (x - 1)

Again, dividing the leading terms, we get:

lim (x -> -∞) (x/x) = 1

So, y = 1 is the horizontal asymptote.

To find oblique asymptotes, we perform polynomial long division. Dividing x + 1 by x - 1:

```
x + 1
```

x - 1 | x + 1

```
- (x - 1)
__________
2
```

So, the quotient is x + 2 and the remainder is 2. Therefore, the oblique asymptote is y = x + 2.

In summary:

- Vertical asymptote: x = 1
- Horizontal asymptote: y = 1
- Oblique asymptote: y = x + 2

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To find the vertical asymptotes of (y = \frac{x + 1}{x - 1}), we identify the values of (x) that make the denominator (x - 1) equal to zero. This happens when (x = 1). Thus, the vertical asymptote is (x = 1).

To find the horizontal asymptote, we examine the behavior of the function as (x) approaches positive or negative infinity. When (x) approaches positive or negative infinity, the term with the highest degree in the numerator and denominator dominates the function. In this case, both the numerator and denominator have the same degree, which is 1. Therefore, to find the horizontal asymptote, we divide the leading coefficient of the numerator by the leading coefficient of the denominator. Since both leading coefficients are 1, the horizontal asymptote is (y = 1).

To find the oblique asymptote, we perform long division of the numerator by the denominator. After dividing (x + 1) by (x - 1), we obtain (y = 1 + \frac{2}{x - 1}). As (x) approaches positive or negative infinity, the term (\frac{2}{x - 1}) approaches zero. Therefore, the oblique asymptote is (y = 1).

In summary:

- Vertical asymptote: (x = 1)
- Horizontal asymptote: (y = 1)
- Oblique asymptote: (y = 1)

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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