Understanding Horizontal and Vertical Asymptotes: Definitions and Examples

Horizontal Asymptote y=0 and a Vertical Asymptote at x = 0

To understand the concept of horizontal and vertical asymptotes, let’s start with definitions

To understand the concept of horizontal and vertical asymptotes, let’s start with definitions.

A horizontal asymptote is a horizontal line that a function approaches as the input variable (x) goes to positive infinity or negative infinity. It can be represented as y = k, where k is a constant.

A vertical asymptote is a vertical line that a function approaches as the input variable (x) approaches a certain value. It can be represented as x = a, where a is a constant.

In your case, the horizontal asymptote is y = 0, which means that as x approaches positive infinity or negative infinity, the function approaches the x-axis or the line y = 0. This indicates that the values of y are getting closer and closer to 0 as x gets larger or smaller without bound.

The vertical asymptote at x = 0 means that as x approaches 0, the function approaches a vertical line at x = 0. This implies that the function has a discontinuity or a sharp change in behavior at x = 0.

To have a better understanding, let’s consider an example function:

f(x) = (x^2 – 1) / x

To find the horizontal asymptote, we need to evaluate the limit of the function as x approaches positive infinity and negative infinity:

lim(x -> ∞) [(x^2 – 1) / x] = lim(x -> ∞) [x – 1/x]
As x approaches infinity, the term 1/x approaches 0, so we are left with:
lim(x -> ∞) x = ∞

Similarly, when x approaches negative infinity:
lim(x -> -∞) [(x^2 – 1) / x] = lim(x -> -∞) [x – 1/x]
Again, as x approaches negative infinity, the term 1/x approaches 0, so we are left with:
lim(x -> -∞) x = -∞

Since the function approaches positive infinity as x approaches positive infinity and negative infinity as x approaches negative infinity, the horizontal asymptote of this function is y = 0.

For the vertical asymptote, we need to determine where the function becomes undefined. In our example function, the function is undefined when the denominator is equal to 0. So, we need to solve the equation:

x = 0

The solution to this equation is x = 0, which means the function has a vertical asymptote at x = 0.

In conclusion, for the given function:
Horizontal asymptote: y = 0
Vertical asymptote: x = 0

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