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Asymptotes

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Asymptotes

Study Guide

Key Definition

An asymptote to a curve is a straight line which the curve approaches. If we go sufficiently far along the line, the curve becomes arbitrarily close.

Important Notes

If a function has a horizontal asymptote y = L, then as x approaches ±∞, the graph approaches the line y = L; it may approach from above or below depending on the sign of f(x) - L.
If a function has a vertical asymptote x = c, then as x approaches c from the left or right, the graph values go to positive or negative infinity.
Diagonal asymptotes are also possible; for instance, the graph of $y = \frac{1}{x} + x$ has the line $y = x$ as an asymptote.

Mathematical Notation

$\frac{1}{x}$ represents a simple rational function with asymptotes on the x-axis and y-axis

$\frac{1}{x} + x$ represents a rational function with a diagonal asymptote

Remember to use proper notation when solving problems

Why It Works

Because of the properties of limits and infinity, the graph of the function gets closer and closer to the line as x or y approaches infinity, but never actually touches or crosses it.

Remember

The x-axis and y-axis can act as asymptotes for certain functions.

Quick Reference

Horizontal Asymptote:$y = c$

Vertical Asymptote:$x = c$

Understanding Asymptotes

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Watch & Learn

Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

An asymptote is a line that a curve approaches, as it heads towards infinity.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

Which function has the x-axis as an asymptote?

Real-World Problem

Question Exercise

Intermediate

Space Travel Scenario

In the context of space travel, imagine that a spaceship is moving along the curve $y = \frac{1}{x}$. As it goes further into space (as x goes to infinity), does it ever cross the x-axis (the horizontal line y = 0)?

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

What happens to the curve $y = \frac{1}{x} + x$ as x goes to infinity and as x goes to negative infinity?

Challenge Quiz

Single Choice Quiz

Advanced

Which function does not have an asymptote?

Recap

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Review key concepts and takeaways