Limit of a Function

Bridge Course, August 2012

Intuition: Function f assigns some value f(x) to each x. We say that f has limit l at x = p when

f(x) is “close” to l whenever x is “close” to p. This means that as x goes closer to p, f(x) goes closer to l.

                    (a)f(x) = x^2                                                                    (b) f(x) = 1/x^2

Note that limit does not depend on f(p). Also, in general, .

Consider a few examples of limit computation:

         (1)                     (2)                    (3)          

Some useful and well-known limits of functions are:     

Let us consider two function f and g. Let us assume  exist, then we have

For example, ,  but  and  are not defined.

For the following problems, determine if the limit exists and if it does, then compute it.

Additional Reading for HAROLD^[1]

Definition: Consider a function f = f(x) defined over real-line. The limit of f as x approaches p is l if

and only if for every real ϵ > 0, there exists real δ > 0 such that |f(x) − l| < ϵ whenever |x − p| < δ.

Note that δ, in general, depends on ϵ, that is, δ = δ(ϵ). We write this as:

Examples: Application of definition

Example 1: In this example, we use the definition of limit of a function to verify .

There are two parts of this proof. We need to show that

(a) Given ϵ > 0, there exists δ > 0 such that

Proof: We have 

Therefore, we let .

(b) Given δ > 0, there exists ϵ > 0 such that

Proof: We have

Therefore, we let .

Example 2: In this example, we use the definition of limit of a function to verify . There are two parts of this proof. We need to show that

(a) Given ϵ > 0, there exists δ > 0 such that

Proof: We have
                                                        

Therefore, we let .

(b) Given ϵ > 0, there exists δ > 0 such that

Prove yourself.

For the following problems, verify the limits using the definition of the limit of a function.