HOW TO EVALUATE LIMITS FROM A GRAPH

Let f be a function of a real variable x. Let a, L be two fixed real values. When x approaches the real value a, f(x) approaches the real value L, we say L is the limit of the function f(x) as x tends to a.

This is written as

                                       lim   f(x) = L
                                    x--> a

Here, we may have a question. When x approachs a, whether x approaches a from its left side or right side.

If x approaches a from its left side, it is called left-sided limit and if x approaches a from its right side, it is called right sided limit.

Left-Sided Limit

For example, let x approach 0 from its left side. Then, x will take the values which are very close to zero, finally it will reach 0 as shown in the picture below.

In the graph shown below, when x approaches a real value a from its left side, f(x) approaches the real value L.

Therefore, L is the left sided limit of the function f(x) as x tends to a.

This is written as

                                       lim   f(x) = L
                                    x--> a^-

In the above limit, to indicate the left-sided limit, there is a negative sign on the right top corner of a.

Right-Sided Limit

Let x approach 0 from its right side. Then, x will take the values which are very close to zero, finally it will reach 0 as shown in the picture below.

In the graph shown below, when x approaches a real value a from its right side, f(x) approaches the real value L.

Therefore, L is the right sided limit of the function f(x) as x tends to a.

This is written as

                                       lim   f(x) = L
                                    x--> a⁺

In the above limit, to indicate the right-sided limit, there is a positive sign on the right top corner of a.

Two-Sided Limit

Consider the left-sided limit and right sided limit are shown below. 

When x approaches the real value a from both its left side and right side, f(x) approaches the same real value L in both the cases. Therefore, L is the two sided limit of the function f(x) as x tends to a.

This is written as

                                       lim   f(x) = L
                                    x--> a

When we write two-sided limit, there won't be any sign on the right top corner of a.

It has been illustrated in the graph shown below.

What if the left-sided limit and right-sided limit are not equal?

Consider the graph shown below.

In the graph shown above, when x approaches the real value a from its left side, f(x) approaches the real value 2.

                                       lim   f(x) = 2
                                    x--> a^-

When x approaches the real value a from its right side, f(x) approaches the real value 3.

                                       lim   f(x) = 3
                                    x--> a⁺

So, the left-sided limit of f(x) and the right sided limit of f(x) are not equal, as x approches a.

                             lim   f(x)   ≠   lim   f(x)
                             x--> a^-             x--> a⁺

If left-sided limit and right-sided limit are not equal, then the two-sided limit does not exist.

That is,

                           lim   f(x) does not exist
                        x--> a

Solved Examples

From the graph shown above, evaluate each of the following limits.

Example 1 :

                                       lim   f(x)
                                    x--> -4^-

Solution :

We have to evaluate left-sided limit of f(x) as x approaches -4.

In the graph shown above, when x approches -4 from its left side, f(x) approaches 1.

Therefore,

                                     lim   f(x) = 1
                                   x--> -4^-

Example 2 :

                                      lim   f(x)
                                    x--> -4⁺

Solution :

We have to evaluate right-sided limit of f(x) as x approaches -4.

In the graph shown above, when x approches -4 from its right side, f(x) approaches -1.

Therefore,

                                     lim   f(x) = -1
                                   x--> -4^-

Example 3 :

                                     lim   f(x)
                                   x--> -4

Solution :

We have to evaluate two-sided limit of f(x) as x approaches -4.

From the solutions of Examples 1 and 2 above,

                           lim   f(x)   ≠   lim   f(x)
                           x--> -4^-           x--> -4⁺

Since, the left-sided limit and right sided limit are not equal, two-sided limit does not exist.

That is,

                           lim   f(x) does not exist
                        x--> -4

Example 4 :

                                      lim   f(x)
                                    x--> 1^-

Solution :

We have to evaluate left-sided limit of f(x) as x approaches 1.

In the graph shown above, when x approches 1 from its left side, f(x) approaches 4.

Therefore,

                                     lim   f(x) = 4
                                   x--> 1^-

Example 5 :

                                      lim   f(x)
                                    x--> 1⁺

Solution :

We have to evaluate right-sided limit of f(x) as x approaches 1.

In the graph shown above, when x approches 1 from its right side, f(x) approaches 4.

Therefore,

                                     lim   f(x) = 4
                                   x--> 1⁺

Example 6 :

                                      lim   f(x)
                                    x--> 1

Solution :

We have to evaluate two-sided limit of f(x) as x approaches 1.

From the solutions of Examples 4 and 5 above,

                           lim   f(x)   =   lim   f(x)
                           x--> 1^-              x--> 1⁺

Since, the left-sided limit and right sided limit are equal, two-sided limit exists.

That is,

                                     lim   f(x) = 4
                                   x--> 1

Example 7 :

                                      lim   f(x)
                                    x--> 2^-

Solution :

We have to evaluate left-sided limit of f(x) as x approaches 2.

In the graph shown above, when x approches 2 from its left side, f(x) approaches 3.

Therefore,

Therefore,

                                     lim   f(x) = 3
                                   x--> 2^-

Example 8 :

                                      lim   f(x)
                                    x--> 2⁺

Solution :

We have to evaluate right-sided limit of f(x) as x approaches 2.

In the graph shown above, when x approches 2 from its right side, f(x) approaches 3.

Therefore,

                                     lim   f(x) = 3
                                   x--> 2⁺

Example 9 :

                                      lim   f(x)
                                    x--> 2

Solution :

We have to evaluate two-sided limit of f(x) as x approaches 2.

From the solutions of Examples 7 and 8 above,

                           lim   f(x)   =   lim   f(x)
                           x--> 2^-             x--> 2⁺

Since, the left-sided limit and right sided limit are equal, two-sided limit exists.

That is,

                                     lim   f(x) = 3
                                   x--> 2

Example 10 :

                                      lim   f(x)
                                    x--> 4^-

Solution :

We have to evaluate left-sided limit of f(x) as x approaches 4.

In the graph shown above, when x approches 4 from its left side, f(x) approaches +∞.

Therefore,

                                     lim   f(x) = +∞
                                   x--> 4^-

Example 11 :

                                      lim   f(x)
                                    x--> 4⁺

Solution :

We have to evaluate right-sided limit of f(x) as x approaches 4.

In the graph shown above, when x approches 4 from its right side, f(x) approaches +∞.

Therefore,

                                     lim   f(x) = +∞
                                   x--> 4⁺

Example 12 :

                                      lim   f(x)
                                    x--> 4

Solution :

We have to evaluate two-sided limit of f(x) as x approaches 4.

From the solutions of Examples 10 and 11 above,

                           lim   f(x)   =   lim   f(x)
                           x--> 4^-             x--> 4⁺

Since, the left-sided limit and right sided limit are equal, two-sided limit exists.

That is,

                                     lim   f(x) = +∞
                                   x--> 4

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