One Sided Limits
Limit of a function is given by lim (x->a) f(x) = L, that is, as x approaches ‘a’ from the right hand side of a, f(x) = L and also, as x approaches a from the left hand side of a, f(x)=L.
One Sided Limits Examples
Let us consider a function f(x) = -x^2+6 for 0<=x<2 and f(x)=x-1 for 2<= x <= 4. When we graph this function, we can see that the function is discontinuous at x=2, and the two pieces of the function approach different values, we get, lim (x->2)f(x) is undefined. We can say that the limit of f(x) as x approaches 2 from the left is 2, and the limit of f(x) as x approaches from the right is 1. We can write it as, lim (x->2-) f(x) = 2 and lim (x->2+) f(x) = 1. The plus sign indicates ‘from the right side’ and the minus sign indicates ‘from the left side’.
There are two one sided limits, Right and left hand limits. Right hand limit which is defined as lim (x->a+) f(x) = L , if f(x) is as close to L as we want for all x sufficiently closer to a and x is greater than a, without actually letting x equal to a. Left Hand Limit is defined as lim(x->a-)=L, if f(x) is as close to L as we want for all x sufficiently closer to a and x is less than a, without actually letting x equal to a.
Finding One Sided Limits Algebraically
Let us consider a piecewise function f(x)= x+2 for x less than or equal to 6 and f(x) = x^2-1 for x greater than 6. The steps involved in finding one sided limits of the given piecewise function are as follows:
Left Handed limit of f(x) is given by lim (x->6-)f(x), all the values of x on the number line lie to the left of 6 which means x is less than 6 and hence we consider the function f(x) = x+2. We plug in the value x=6 in the function which gives, 6+2 = 8
Right Handed Limit of f(x) is given by lim(x->6+) f(x), all the x values considered on the number line lie to the right of 6, so, x is greater than 6 and hence we consider the function f(x) = x^2-1. We shall plug in the value x=6 in the function which gives, (6)^2-1 = 36-1=35
Lim(x->6)f(x) does not exist as the right handed limit is not equal to the left handed limit
