Limit, continuity and differentiability

Estimating a limit using a graph

Now that we know how to identify a limit in a simple equation, let’s see how to identify a limit. 

Although the equation looks different than the one we initially saw, the fundamental concept remains the same. 

X is the variable, f(x) is the limit.

Let’s look at what the graph equation is supposed to represent. 

How did we arrive at this point on the graph, though?

Let’s try to understand what the graph represents visually.

1. The graph runs through 2 for the value of +x at all instances on x on the graph.
2. The graph coincides at one point, which is 0=3 for all values of x approaching 3. This will hold true for ANY value for the limit as f(3)=0 has no value on the limit of x as it approaches 3.
3. We know the limit for f(x)=2  for all instances, except for when x=3, so we can safely assume that the limit, f(x), is 2. 

What is continuity?

Qualitatively the graph of a function is said to be continuous at x = d if travelling along with the graph of the function and crossing over the point at x = d either from left to right or from right to left, and one does not have to lift his pen. If one has to lift his pen, the function’s graph breaks or gets discontinued at x = d.

A function f(x) is continuous at x = d, if

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