# What is a Limit?

*One of the first concepts learned in Calculus is also one of the more confusing ones*

Simply put, a limit is the value a function takes as x approaches some number (usually a). In the image above, x approaches 2. The word “approaches” is very important because x can never equal the limit. This subtlety in language can actually be used for proving that the derivative of a constant is zero (more on derivatives later).

**Why do we use limits and not just regular functions?**

There are a few reasons why. One of them is that it almost forces people to think in terms of variables rather than values. A second reason is that it tells us more about a function that is undefined at a specific point. Referring to the above image, if we were asked to find f(2), the value would be undefined. However, we can find the value of the limit as x approaches 2 from the left. Any guesses to what it could be? In this case, the limit as x approaches 2 from the left is different from the limit as x approaches 2 from the right, because the function is discontinuous.

So, limits just helped us find out three things about a weird function:

1) Where the function is discontinuous

2) What kind of discontinuity the function has

3) Two probable values that f(2) could take if it was… more continuous.

**Why are limits useful?**

Limits of a function are useful because they are not the same as the values of a function. This allows us to do neat things such as divide by zero. In fact, when solving for the slope of a tangent line to a graph (a derivative) we must take the very familiar slope formula and evaluate it as the denominator approaches zero. That is all a derivative is: the slope of a tangent line at a specific point on a graph. However, it has many implications in real life too, such as the speedometer on a car. These derivatives would not exist without limits.

Image Source: http://en.wikipedia.org/wiki/Limit_%28mathematics%29