09. Limits(Intuitive Introduction)

Dated: 30-10-2024

Tangent Problem

Given a graph and a point, find tangent to the graph at that point.

The problem gave rise to differential calculus.

Area Problem

Given a graph, find area under it in interval¹ \([a, b]\).
This problem gave rise to integral calculus.

Area as a Limit

We can find the area under a curve by dividing it into smaller and smaller rectangles.
And then taking the sum of the area of these rectangles, we get our area.

Limits

Imagine a function²

\[f(x) = \frac{\sin(x)}{x}\]

Notice how it is undefined at \(x = 0\).
This is where the Limit comes into play.

\[\frac{\sin(x)}{x} = undefined\]

To check if a limit exists, meaning what does the function² approaches, we check from both left and right directions on the x-axis.

From Right

What does the function² approaches as \(x\) approach to 0 from right direction.

\[\lim_{x \rightarrow 0^+}\frac{\sin(x)}{x} = 1\]

From Left

What does the function² approaches as \(x\) approach to 0 from left direction.

\[\lim_{x \rightarrow 0^-}\frac{\sin(x)}{x} = 1\]

Therefore, to check the existence of lim,

\[\lim_{x \rightarrow 0^-}\frac{\sin(x)}{x} = \lim_{x \rightarrow 0^+}\frac{\sin(x)}{x} = 1\]