What is limit?

In mathematics, a limit is a fundamental concept used to describe the behavior of a function as its input values approach a specific point. The idea is to determine what happens to the function’s values as you get closer and closer to a particular value without necessarily reaching it. Let’s explain this concept with an example:

Consider the function f(x) = 1/x. We want to find the limit of this function as x approaches the value 0. In mathematical notation, we write this as:

lim (x → 0) (1/x)

This means we’re interested in understanding what happens to the function as x gets closer and closer to 0.

Now, if we directly substitute x = 0 into the function, we get:

f(0) = 1/0

However, dividing by zero is undefined in mathematics, so we can’t determine the value of the function at x = 0. This is where limits come into play.

To find the limit, we examine what happens as x approaches 0 from both sides, which means as x gets closer to 0 while still being positive (right-hand limit) and as x gets closer to 0 while still being negative (left-hand limit).

Let’s consider values of x that are close to 0 but not equal to 0:

  • When x is 0.1 (right-hand side): f(0.1) = 1/0.1 = 10
  • When x is -0.1 (left-hand side): f(-0.1) = 1/(-0.1) = -10

As we get closer and closer to x = 0 from both sides:

  • The right-hand limit, lim (x → 0⁺) (1/x), approaches positive infinity (i.e., the values get larger and larger).
  • The left-hand limit, lim (x → 0⁻) (1/x), approaches negative infinity (i.e., the values get smaller and smaller).

In this case, the function does not have a finite limit as x approaches 0 because it behaves differently from the left and right sides. Instead, we say that the limit is “undefined” or “does not exist” because it doesn’t settle on a specific value.

So, for the function f(x) = 1/x as x approaches 0:

lim (x → 0) (1/x) is undefined or does not exist.

This example illustrates how limits help us analyze and understand the behavior of functions as they approach certain values or points of interest.

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