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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.

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