One sided limits in calculus

Calculus defines one sided limits as one of the two limits of the function of x as x tries to approach a given point either from the left or from the right. The one sided limits calculus deals with different values of function for both methods.

Ways for finding one sided limits:

The right hand one sided limit is symbolized by right hand one sided limitwhile that for left hand one is symbolized by left hand one sided limit

The steps for finding one sided limits are given below.

  • For right hand limit, replace x by a – h. It makes x -> a- equal to h -> 0. Thus the equation becomes lim h->0 f(a – h).

  • For the left hand limit, we will replace x by a + h. It makes x -> a+ equal to h -> 0. Thus the equation becomes lim h->0 f(a + h).

  • The above equations can be evaluated by putting the value of function in it to give respective sided limits.

Substituting the values of limit points in the equation will help finding one sided limits algebraically. We will see those one sided limits problems in the below section.

Solving some of the one sided limits examples:

Lets solve some of the one sided limit examples to have a better understanding of the topic.

(1) Does the overall limit exist for f(x) = √x for x -> 0:

If the function tries to reach 0 from right side, it is defined for all the values. But if the function approaches from the left side, the function will have complex values which are not allowed. Hence overall limit does not exist for this function. Only right side limits exist.

(2) Find overall limit for g(x) = x3 + 1, x < 0 and g(x) = x + 1, x ≧ 0:

Left hand limit for g(x) will be L = 03 + 1 = 1. Now right hand limit will be L = 0 + 1 = 1. Since both the limits are same, overall limit exists for g(x) and is 1.

Thus we have evaluated one sided limits examples and learned calculating the overall limit from them.

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