5.10 Limits  (Page 2/5)

Left hand limit or left limit

Left hand limit is an estimate of function value from a close point on the left of test point. It answer : what would be function value – not what is - at the test point as we approach to it from left? Symbolically, we represent this limit by putting a “minus” sign following test point “a” as “a-“.

$$\underset{x\to a-}{\overset{}{\lim }}f\left(x\right)=L_l$$

In terms of delta – epsilon definition, we write :

$$L_l-\delta <f\left(x\right)<L_l+\delta \text{for all x in}a-\in <x<a$$

Graphically, we represent left limit by a curve which points towards limiting value from left terminating with an empty small circle at the test point. The empty circle denotes the limiting value. Since it is an estimate based on nature of graph – not actual function value, it is shown empty. In case, function value is equal to left limit, then circle is filled. If limit approaches infinity, then we show a graph with out terminating circle, approaching an asymptote towards either positive or negative infinity.

Right hand limit or right limit

Right hand limit is an estimate of function value from a close point on right of test point. It asnwers : what would be function value – not what is - at the test point as we approach to it from right? Symbolically, we represent this limit by putting a “plus” sign following test point “a” as “a+“.

$$\underset{x\to a+}{\overset{}{\lim }}f\left(x\right)=L_r$$

In terms of delta – epsilon definition, we write :

$$L_l-\delta <f\left(x\right)<L_l+\delta \text{for all x in}a<x<a+\in$$

Graphically, we represent right limit by a curve which points towards limiting value from right terminating with an empty small circle at the test point. If limit approaches infinity, then we show a graph with out terminating circle, approaching an asymptote towards either positive or negative infinity.

Limit at a point

Limit is an estimate of function value from close points from either side of test point. If left and right limits approach same limiting value, then limit at the point exists and is equal to the common value. Clearly, if left and right limits are not equal, then we can not assign an unique value to the estimate. Clearly, limit of a function answers : what would be function value – not what is - at the test point as we approach to it from either direction? Symbolically, we represent this limit as :

$$\underset{x\to a}{\overset{}{\lim }}f\left(x\right)=L_l=L_r=L$$

In terms of delta – epsilon definition, we write :

$$L-\delta <f\left(x\right)<L+\delta \text{for all x in}a-\in <x<a+\in$$

Graphically, we represent the limit by a pair of curves which point towards limiting value from left and right terminating with a common empty small circle at the test point. If limit approaches infinity, then we show a graph with out terminating circle, approaching an asymptote from either direction in the direction of either positive or negative infinity.

Limit and continuity

It has been emphasized that limit is an estimate of function value based on function rule at a point. This estimate is not function value. Function value is defined by the definition of function at that point. However, if function is continuous from the neighboring point to the test point, then limit should be equal to function value as well. Consider modulus function :