Intuitive concept of limit

The concept of the limit of any function is the basis of all calculus. It is used to define derivation and integration, which form the basis of calculus. Consider a line on a graph where y= f(x). Here, when f(x) goes closer to a particular number L, x gets closer and closer to c from either side of the line, then L is the limit of f(x) as x approaches c. The behavior is expressed by writing:

$\lim _{x\to c}f(x)=L$

Intuitive idea of limits

As the variable x approaches an on the graph above, the function f(x) approaches L.The limit of an actual function may look like the figure below where f (x) equals x2. As “x” approaches 1, f(x) approaches 1.

To express the function above we write:
$\lim _{x\to 1}x^{2}=1$

One-sided limits:

From the graph above, as you go from left to right and get closer to point a, notice that $\lim _{x\to a^{-}}f(x)=1$

Since y value is moving closer to 1 as x value is going closer to a from the left.

In general terms, the left-hand limit of f(x) as x approaches a from the left can be expressed as $\lim _{x\to a^{-}}f(x)=L$

We can assume the values of f(x) arbitrarily close to L by stating x < a. From the graph above, as you go from right to left and move closer to point a, notice that $\lim _{x\to a^{+}}f(x)=3$

Since y values are going closer to 3 as x value is going close to a from the right.

In general terms, the right-hand limit of f(x) as x approaches a from the right can be expressed as $\lim _{x\to a^{+}}f(x)=L$

We can assume the values of f(x) arbitrarily close to L by stating x > a.

Theorem: The 2­ sided limit $\lim _{x\to a}f(x)=L$ exists if and only if the $\lim _{x\to a^{-}}f(x)=L$ and $\lim _{x\to a^{+}}f(x)=L$ exist and $\lim _{x\to a}f(x)=L$iff$\lim _{x\to a^{-}}f(x)=L$ and $\lim _{x\to a^{+}}f(x)=L$

All of the graphs below have the same one-sided limits and none of the two-sided limits exist. It does not matter what happens right at the value when determining limits.

Sometimes 1­ sided or 2­ sided limits fail to exist because the values of the function increase or decrease without bound. The +ive and -ive infinity (∞) are not real numbers, they describe specific ways in which the limits fail to exist.

Follow the graph above from left to right. As the x values get closer to 0, notice that: $\lim _{x\to 0^{-}}f(x)=-\infty $

Since y values are decreasing without abounding as x goes close to 0 from the left.

Follow the graph from right to left. As the x values get closer to 0, notice that: $\lim _{x\to 0^{+}}f(x)=+\infty $

Since y values are increasing without bound as x goes closer to 0 from the right.

 

 

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