Basic limit graph

Some situations where you might be asked to find the limit!

 How to read: lim as x approaches a of f(x) is equal L

In i) a is in the domain of the function f and lim as x approaches a is equal to L ( where L also happens to be f(a)).
In ii) a is in not in the domain of a function f and limit as x approaches a is equal to L (where L is not equal to f(a)).

In iii) a is not in the domain (the a E Df above is wrong). Limit as x approaches a from left side of the function f(x)= L1 whereas limit as x approaches a from right side is equal to L2. Since, L1 is not equal to L2, limit as x approaches a of f(x) does not exist.

So, today we learned in order for limit to exist the left side and the right side limit must equal to each other. The limit may also exist even if the point is not in the domain (like in ii).

More Basic Limits that you may encounter

This is a follow-up for previous posts with an example

In iv) a is not in the domain of the function. Limit as x approaches a does not exist because left hand side limit is not equal to right hand side limit.
In v) a is in the domain although there is a vertical asymptote shown in the graph. Limit as x approaches a from left side of f(x) is equal to L1(some number) and limit as x aprroaches a from right of f(x) is equal to infinity. Hence, limit as x approaches a does not exist. However, the limit as x approaches a from left side exist.

In 6) a is not in the domain because there is a vertical asymptote in the graph. However, more accurate answer to why a is not in the domain is: because limit from left hand side goes to (+)infinity and limit as x approaches a from right hand side goes to negative infinity.
In 7) a is not in the domain. Limit as x approaches a from left side is (-) infinity and limit as x approaches a from right side is also (-) infinity. Hence, limit as x approaches a is (-) infinity.

in Example (E): f(x) = ln x  

domain: (0, ∞) all numbers greater than 0

lim as x approaches 0 from right is (-) infinity

limit as x approaches 0 from right is not possible because it is out of the domain.

since limit as x approaches 0 from right of f(x) is (-) infinity, at x=0 there exists a vertical asymptote.

Limit problems and limit when y is equal to constant number

Graph of trigonometric function

(These examples are not made or owned in anyway by me)

It is the graph of trigonometric function so it should be all curve (it looks like a line in the picture but it is a curve)

y = c is just a horizontal line; Hence, limit as x approaches any number of f(x) is always C.

when y=x, then the limit as x approaches any number (a) of f(x) is always a.

1) when there is a constant multiplying the variable you can take the number out in front and then take the limit of the variable. Its one of the rules that comes in handy later on when you take limits of huge functions.

2) I think it is pretty self explanatory. If not see example.