## The definition of the limit

The limit of a function f(x), at some value a, is simply, what the value of the function approaches when x gets close to a.

Of course, most of the times, the limit is equal to the value of f(a). (What else would you expect?) Most of the times. There are exceptions. Those exceptions are what make the concept of the limit tricky and time-consuming. Let's take a moment to think what kind of function would it take to make the limit different from the value of the function.

Of course, polynomial functions who are defined everywhere, like y=x, are easy prey to our mathematical minds. The limits of those functions at any x is equal to the value of the function at that point.

Right. Here, I present a part of the

1. Functions who are not defined at a. There is no value at a to talk of.

(e.g. Rational functions with the denominator as zero at the point x=a.)

2. Rational functions whose numerator or denominator appears to march off to infinity. Infinity allows no limit.

3. Functions who vibrate violently around the value a.

4. Functions who approach different values when x gets close to a from the left side and the right side.

Be extra careful when encountered with such functions.

Of course, most of the times, the limit is equal to the value of f(a). (What else would you expect?) Most of the times. There are exceptions. Those exceptions are what make the concept of the limit tricky and time-consuming. Let's take a moment to think what kind of function would it take to make the limit different from the value of the function.

Of course, polynomial functions who are defined everywhere, like y=x, are easy prey to our mathematical minds. The limits of those functions at any x is equal to the value of the function at that point.

Right. Here, I present a part of the

__FUNCTION BLACKLIST__, whose contents produce messy limits, but sadly, takes up most of the space in calculus tests.1. Functions who are not defined at a. There is no value at a to talk of.

(e.g. Rational functions with the denominator as zero at the point x=a.)

2. Rational functions whose numerator or denominator appears to march off to infinity. Infinity allows no limit.

3. Functions who vibrate violently around the value a.

4. Functions who approach different values when x gets close to a from the left side and the right side.

Be extra careful when encountered with such functions.

__The limit might not appear to be what it is.__