Limits at Infinity and Horizontal Asymptotes

1. The Concept of a Limit at Infinity

Let's start with a simple, intuitive idea. Consider the function $f(x)=\frac{1}{x}$. What happens to the value of $f(x)$ as $x$ gets bigger and bigger?

• If $x=100$, $f(x)=0.01$
• If $x=10,000$, $f(x)=0.0001$
• If $x=1,000,000$, $f(x)=0.000001$

As $x$ increases, $f(x)$ gets closer and closer to 0. We say that the limit of $f(x)$ as $x$ approaches infinity is 0, and we write this as:
${\lim }_{x\to \infty }\frac{1}{x}=0$
The same thing happens as $x$ becomes a very large negative number:
${\lim }_{x\to -\infty }\frac{1}{x}=0$
Graphically, this means the curve of the function gets closer and closer to the horizontal line $y=0$ (the x-axis) as we look far to the right or far to the left. This line is called a horizontal asymptote.

Let's get a more formal definition of this connection.

Definition: The line $y=L$ is a horizontal asymptote of the graph of a function $f(x)$ if either
${\lim }_{x\to \infty }f(x)=L\text{or}{\lim }_{x\to -\infty }f(x)=L$

2. Evaluating Limits of Rational Functions at Infinity

For a simple function like $1/x$, the limit is obvious. But what about a more complex rational function, like $f(x)=\frac{4x^2+5}{2x^2-1}$?

The key algebraic technique is to divide both the numerator and the denominator by the highest power of $x$ that appears in the denominator. This helps us because it creates many terms of the form $c/x^n$, which we know go to 0 as $x$ approaches infinity.

Let's solve ${\lim }_{x\to \infty }\frac{4x^2+5}{2x^2-1}$:

1. The highest power of $x$ in the denominator is $x^2$.
2. Divide every term in the expression by $x^2$:
   ${\lim }_{x\to \infty }\frac{\frac{4x^2}{x^2}+\frac{5}{x^2}}{\frac{2x^2}{x^2}-\frac{1}{x^2}}={\lim }_{x\to \infty }\frac{4+\frac{5}{x^2}}{2-\frac{1}{x^2}}$
3. Now, we apply the limit. As $x\to \infty$, both $\frac{5}{x^2}$ and $\frac{1}{x^2}$ approach 0.
   $\frac{4+0}{2-0}=\frac{4}{2}=2$
   So, ${\lim }_{x\to \infty }f(x)=2$. This tells us that the function has a horizontal asymptote at $y=2$.

The following video walks through this formal process for different types of rational functions.

The Shortcut: Comparing Degrees

The division method always works, but it leads to a convenient shortcut for rational functions. All you need to do is compare the degree (the highest index or power) of the polynomial in the numerator ($N$) with the degree of the polynomial in the denominator ($D$).

1. If $N<D$ (bottom-heavy): The denominator grows faster than the numerator. The limit is 0.
   • Example: ${\lim }_{x\to \infty }\frac{x+1}{x^2+3}=0$
2. If $N=D$ (balanced): The numerator and denominator grow at the same rate. The limit is the ratio of the leading coefficients.
   • Example: ${\lim }_{x\to \infty }\frac{4x^2+5}{2x^2-1}=\frac{4}{2}=2$
3. If $N>D$ (top-heavy): The numerator grows faster than the denominator. The limit is $\infty$ or $-\infty$ (the limit does not exist as a finite number). There is no horizontal asymptote.
   • Example: ${\lim }_{x\to \infty }\frac{x^3-2}{x^2+1}=\infty$

This next video explains these rules very efficiently.

3. More Complex Cases

Functions with Square Roots

When dealing with square roots, we use the same division technique, but with a twist. To divide a term inside a square root by $x^n$, you must bring it inside the root as $x^{2n}$. For example, $\frac{\sqrt{x^2+1}}{x}=\sqrt{\frac{x^2+1}{x^2}}=\sqrt{1+\frac{1}{x^2}}$.

Things get particularly interesting when $x\to -\infty$. Remember that for negative $x$, $\sqrt{x^2}=|x|=-x$. This sign difference is crucial.

• When $x\to \infty$ (i.e., $x>0$), $x=\sqrt{x^2}$.
• When $x\to -\infty$ (i.e., $x<0$), $x=-\sqrt{x^2}$.

This can lead to a function having two different horizontal asymptotes: one for $x\to \infty$ and another for $x\to -\infty$.

Other Function Types

• Exponential Functions: The limit of $e^x$ as $x\to \infty$ is $\infty$, but as $x\to -\infty$ it's 0. The opposite is true for $e^{-x}$.
• Trigonometric Functions: Functions like $\sin (x)$ and $\cos (x)$ oscillate and do not approach any single value as $x\to \infty$. Therefore, ${\lim }_{x\to \infty }\sin (x)$ does not exist. However, if it's part of a fraction like $\frac{\sin (x)}{x}$, the limit becomes 0. This is because the numerator is "squeezed" or "sandwiched" between -1 and 1 while the denominator grows to infinity.

Conclusion

Today, we've extended our understanding of limits to encompass the infinite. By analysing the end behaviour of functions, we can determine their long-term trends and identify the horizontal asymptotes that guide their graphs.

Key Takeaways:

• Limits at infinity describe the value a function approaches as $x$ grows infinitely large or small.
• If ${\lim }_{x\to \pm \infty }f(x)=L$, then the line $y=L$ is a horizontal asymptote.
• For rational functions, the limit at infinity is determined by comparing the degrees of the numerator and denominator.
• The shortcut for rational functions is:
  • Bottom-heavy: Limit is 0.
  • Balanced: Limit is the ratio of leading coefficients.
  • Top-heavy: Limit is $\infty$ or $-\infty$ (no horizontal asymptote).
• Be especially careful with functions involving square roots when evaluating limits at $x\to -\infty$, as the sign can change.

In our next lesson, we will use our complete understanding of limits to define one of the most important concepts in calculus: continuity. We'll finally give a precise mathematical meaning to the idea of a function whose graph can be drawn without lifting your pen from the paper.