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Limit of a Function

So, what exactly is a limit of a function?

Imagine you've got a function f(x) and x is approaching a specific value, let's call it x0. The limit is the finite or infinite value that f(x) gets closer and closer to as x inches its way toward x0.

Keep in mind, x0 isn't just any value – it's a specific point within the function's domain and could even be infinite (∞).

But what does this all mean, really? To bring it down to earth, let's walk through a couple of practical examples.

A Practical Example

Consider the function f(x)=x+1. Its domain stretches from negative infinity (-∞) to positive infinity (+∞).

the graph of the function x+1

Let's pluck a point from the domain, for instance, x0=2.

What happens to f(x)=x+1 as x sneaks up on x0?

If you approach x0=2 from the right, the function f(x) approaches three f(x)→3.

the right-hand limit of the function as x approaches x0=2 from the right

If you approach x0=2 from the left, the function f(x) also approaches three f(x)→3.

the limit of the function on the left hand

Well, whether you're approaching x0=2 from the right or the left, you'll find that the function f(x) is cozying up to three – in mathematical terms, we say f(x)→3.

What this means is that as the function f(x) nudges closer to the same value (l=3) from both sides, we can confidently say that the limit of the function as x approaches x0=2 is 3.

the limit of the function f(x)=x+1 as x approaches 2 is 3

Keep in mind, this is a simplified example to demystify the concept of a limit. It's worth noting that a function's limit may not always exist at certain points.

Example 2

Take the function f(x)=1/x and compute the limit as x edges toward x0=0.

If you approach x0=0 from the right, the function f(x) tends to positive infinity (+∞).

the limit of the function as x approaches zero from the right is positive infinity

If you approach x0=0 from the left, the function tends to negative infinity (-∞).

the limit as x approaches zero from the left is negative infinity

In this case, the function approaches two different values.

From the right, f(x) skyrockets to positive infinity (+∞), but from the left, it plummets to negative infinity (-∞).

the limit of the function does not exist

What this illustrates is that the function is veering towards two wildly different values as x approaches zero, and therefore, we say the limit of the function as x approaches zero simply does not exist.

Example 3

A limit isn't always bound to be a finite number.

Let's take a look at the function f(x)=1/x2.

the graph of the function f(x)=1/x^2

Then, calculate the limit of the function as x approaches zero (x0=0).

If you approach x0=0 from the right, the function f(x) tends to positive infinity (+∞).

the limit as x approaches zero from the right is positive infinity

If you approach x0=0 from the left, the function f(x) also tends to positive infinity.

the limit of the function as x approaches zero from the left is positive infinity

In this case, as x tiptoes towards zero from either direction, f(x) soars to positive infinity (+∞).

the limit of the function as x approaches zero is positive infinity

So, in this case, regardless of the direction you're coming from, the function is heading towards positive infinity.

Therefore, we can conclude that the limit of the function as x approaches zero is positive infinity.

Now, you might be wondering, what if x0 isn't a finite number? Well, it's absolutely possible to calculate the limit as x approaches infinity, too.

Example 4

Back to our function f(x)=1/x2.

This time, let's compute the limit as x approaches positive infinity.

the limit of the function as x approaches positive infinity is zero

As you get closer and closer to x0=+∞ from the left, the function f(x) eases towards zero.

Since positive infinity (+∞) is not a number but a symbol, you can only approach it from one direction – the left.

Therefore, we can safely conclude that the limit of the function f(x) as x approaches +∞ is indeed zero.

Remember, it's also possible to compute the limit as x approaches negative infinity. While in this case the limit also happens to be zero, this is by no means a rule. It might just as well not exist or be an entirely different value.
the limit as x approaches negative infinity is zero

By now, you should have a solid grasp on what function limits are and why they're important.

With this hands-on overview under your belt, you're ready to dive deeper into the formal definition of a limit and understand it even more profoundly.




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