Limit of a function is denoted as :

lim x → a f x = L lim x → a f x = L

We should read this notation carefully. It is the “limit of function” which is "equal to" L - not the function value. As far as function is concerned it is approaching “L” and value of function, f(a), may or may not be equal to “L”.

Nature of function is not known by its value at a point. Rather, it is known by the value it is likely to have at a neighboring point. Here, we consider hypothetical set up in order to understand the concept. Our job is to find the approaching value which the function will have and which can be represented as “L”. This we do by determining function value a little to the right towards test point if we approach the test point from left. Similarly, we approach the test point from right by determining function value a little to the left towards test point. If we approach the same value from either side from a very close point, then we say that limit of function at test point is “L”.

Important to note here is that this approaching value of function, L, at the test point, x=a, may or may not be the function value f(a). We should understand that the mechanism of piece-wise function definition allows us to define any function value for any point in the domain of function. Further, if test point is a singularity of domain (a point where function is not defined), then there is no function value at the test point.

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-“.

lim x → a − f x = L l lim x → a − f x = L l

In terms of delta – epsilon definition, we write :

L l - δ < f x < L l + δ for all x in a − ∈ < x < a L l - δ < f x < L l + δ for all x in a − ∈ < x < a

Figure 1: Left hand limit

Left hand limit

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 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+“.

lim x → a + f x = L r lim x → a + f x = L r

In terms of delta – epsilon definition, we write :

L l - δ < f x < L l + δ for all x in a < x < a + ∈ L l - δ < f x < L l + δ for all x in a < x < a + ∈

Figure 2: Right hand limit

Right hand limit

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 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 :

lim x → a f x = L l = L r = L lim x → a f x = L l = L r = L

In terms of delta – epsilon definition, we write :

L - δ < f x < L + δ for all x in a - ∈ < x < a + ∈ L - δ < f x < L + δ for all x in a - ∈ < x < a + ∈

Figure 3: Limit at a point

Limit at a point

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.

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 :

       |  x  ; x>0
f(x) = |  0  ; x=0
       | -x  ; x<0

Figure 4: Modulus function

Modulus function

Clearly, at test point x=0,

L = f 0 = 0 L = f 0 = 0

This is an important result which gives us a method to determine limit of a function. If function is continuous, simply put the test value into function definition. The value of function is limit of function at that point.

If function rule changes exactly at the test point, then limit of the function, L, and value of function, f(a), are not same. In order to clearly understand the implication of the statement about inequality of limit and function value, we consider a modification to the modulus function :

        |  x;  x>0
f(x) =  |  1;  x=0
        | -x;  x<0

Figure 5: Modified modulus function

Modified modulus function

At test point x=0,

L = 0 L = 0 f 0 = 1 f 0 = 1 L ≠ f 0 L ≠ f 0

Therefore, limit of function exists at x=0 even though it is not equal to function value. This is an important result which gives us a method to determine limit of a piece-wise defined functions. We need to evaluate function from both left and right side. If limits are equal from both sides, then limit of function at test point is equal to either limit. However, if left and right limits are not equal then limit of function does not exist at the test point.

Singularity or exception point is a point where function is not defined. It is outside definition of function. However, function can point (or tend or approach) to a value at a point where it is not defined. Limit as we know estimate value from a close point where function exits and can project a value based on function definition at points very close to exception point. Consider limit of a rational function :

lim x → 1 x - 1 x + 3 x - 1 lim x → 1 x - 1 x + 3 x - 1

Figure 6: Rational function

Rational function

The singularity of function is obtained by setting denominator to zero. Thus, singularity exists at x=1. We want to know nature of function about this point. In other words, we want to know what would have been the value of function at this point had the function been defined there. For this, we need to evaluate left and right limit at this point. Graphically, there is a hole in the graph of the function. How can we estimate value of function at a point if it is not defined there ?

We keep linear factor in the denominator to know singularity. Extrapolating value at the singularity is a reverse process. We need to calculate function value in the neighborhood, where function is defined. For this, we require to remove linear factor from the denominator. Canceling out (x-1) from both numerator and denominator, we have :

lim x → 1 x - 1 x + 3 x - 1 = lim x → 1 x + 3 = 4 lim x → 1 x - 1 x + 3 x - 1 = lim x → 1 x + 3 = 4

Thus, function is not defined at x=1, but its limit at the point is 4. This means that nature of function in its immediate vicinity is such that the function should have attained a value of 4 had it been estimated on the basis of nature of function in the neighborhood.

This is again an important result which gives us a method to determine limit of a function, when function is not defined at certain point or has other indeterminate or meaningless forms. We need to simplify function expression till we get a form which can be evaluated.

Let us now consider reciprocal function :

f x = 1 x f x = 1 x

Its singularity is obtained by setting denominator to zero. Thus, singularity exists x=0 for the reciprocal function. As such, domain of function is R-{0}. In order to know the function, we need to know nature of function in the vicinity of undefined point. We can do this by evaluating limit on either side of the singularity.

Figure 7: Reciprocal function

Reciprocal function

lim x → 0 − 1 x = − ∞ lim x → 0 − 1 x = − ∞

lim x → 0 + 1 x = ∞ lim x → 0 + 1 x = ∞

Important point to underscore here is that limiting values of x or f(x) as infinity is a valid estimates. To be more explicit, value of function can approach infinity as limiting value. In this case, left and right limits are not same. Therefore, limit of function does not exist at exception point x=0. In order to explore limit at exception point, we consider the case of modulus of reciprocal function. In this case also, function is not defined at x=0. But, for x<0; |x| = -x and for x>0; |x| = x. Hence, left and right limits are :

lim x → 0 − | 1 x | = lim x → 0 − − 1 x = ∞ lim x → 0 − | 1 x | = lim x → 0 − − 1 x = ∞

lim x → 0 + | 1 x | = lim x → 0 + 1 x = ∞ lim x → 0 + | 1 x | = lim x → 0 + 1 x = ∞

Figure 8: Modulus of reciprocal function

Modulus of reciprocal function

In this case, left and right limits are equal. Therefore, limit of function exist at exception point x=0 and it is given as :

lim x → 0 | 1 x | = ∞ lim x → 0 | 1 x | = ∞