Asymptotes are important for understanding how graphs of functions act, especially when it comes to limits. They show us where a function doesn’t behave normally, often leading to very large numbers (like infinity) or values that cannot be defined.

Types of Asymptotes:

1. Vertical Asymptotes: These happen when a function goes towards infinity as the value of (x) gets close to a certain number. For example, in the function (f(x) = \frac{1}{x-2}), there is a vertical asymptote at (x = 2). This means that as (x) gets very close to 2 from the left, (f(x)) goes down to negative infinity, and as (x) gets very close to 2 from the right, (f(x)) goes up to positive infinity.

2. Horizontal Asymptotes: These show what happens to a function as (x) goes to infinity (very large numbers). For example, the function (f(x) = \frac{2x+3}{x+1}) has a horizontal asymptote at (y = 2) because as (x) gets really large, (f(x)) approaches 2.

Connection to Limits:

• Graph Interpretation: Asymptotes help us see the limits of a function when looking at its graph.
• Behavior at Infinity: Horizontal asymptotes tell us the value a function is heading towards as (x) becomes very large or very small. Vertical asymptotes point out the values that the function cannot reach.

In short, asymptotes show us the limits of functions. They help us understand how functions behave near points where they become undefined or extreme.