To understand how functions behave at their ends, we need to learn about limits. Limits help us see what happens to functions as they get really large or really small. This is important for looking at how functions act in the long run, especially at infinity or near vertical asymptotes.

What are Limits at Infinity?

Limits at infinity show us what happens to a function when the input values (like $x$) grow larger and larger or go towards negative infinity.

For example, if we have a function $f(x)$, calculating the limit as $x$ approaches infinity means we want to find out what value $f(x)$ gets closer to as $x$ gets bigger and bigger. We write this as:

$$\lim_{x \to \infty} f(x)$$

Let’s look at a few types of functions to see how this works:

1. Linear Functions:
   Take $f(x) = 2x + 3$.

   As $x$ goes to infinity, we notice that the $2x$ part is the biggest. So,

   $$\lim_{x \to \infty} (2x + 3) = \infty$$

   This means the function keeps growing as we go to the right on the graph.

2. Quadratic Functions:
   Now consider $g(x) = x^2 - 4x + 1$.

   Here, we also find that:

   $$\lim_{x \to \infty} (x^2 - 4x + 1) = \infty$$

   The $x^2$ part is in charge again, showing that this function also increases infinitely as $x$ goes up.

3. Rational Functions:
   Let’s check a rational function like $h(x) = \frac{3x^2 + 2}{2x^2 + 1}$.

   As $x$ approaches infinity, we find that:

   $$\lim_{x \to \infty} \frac{3x^2 + 2}{2x^2 + 1} = \frac{3}{2}$$

   Here, since the top and bottom have the same highest degree, the function levels off at $1.5$.

Now we’ve seen that some functions grow forever and others settle at a specific number as $x$ goes to infinity. We can do the same kind of analysis when $x$ goes to negative infinity:

$$\lim_{x \to -\infty} f(x)$$

What are Vertical Asymptotes?

Vertical asymptotes happen when we see what a function does as $x$ approaches a certain number. These points often occur when the function gets really big, often because we are dividing by zero.

For example, take the function $j(x) = \frac{1}{x - 3}$. When $x$ gets close to 3, the bottom part gets closer to zero:

$$\lim_{x \to 3^+} j(x) = +\infty$$

And conversely,

$$\lim_{x \to 3^-} j(x) = -\infty$$

So, there’s a vertical asymptote at $x = 3$. The function goes up to infinity when getting close to 3 from the right and down to negative infinity when approaching from the left.

In Summary:

Using limits helps us study how functions behave at extreme values and near critical points like vertical asymptotes. Here’s a simple way to remember this:

• Limits at Infinity:

  • Check how functions behave as $x \to \infty$ or $x \to -\infty$.
  • Find out if the function keeps growing, settles on a number, or bounces around.

• Vertical Asymptotes:

  • Look at what happens when functions get close to certain x-values where they might blow up (become undefined).
  • Use limits from both sides to see if they rise to positive or negative infinity.

By understanding these ideas, students build a strong foundation for more complex math topics later on. They learn to visualize what functions are doing as they stretch to extremes. So, limits are not just tools for calculations; they’re essential for understanding how functions work!