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2 limit of a function

Page history last edited by CAI, Sheng 11 years, 10 months ago

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, Formula.

 

Formula

Consider a few examples of limit computation:

 

Formula

 

Formula

 

Formula

 

Formula

 

Formula

 

Formula 

  

         (1) Formula                    (2) Formula                   (3) Formula         

     

Formula

Some useful and well-known limits of functions are:     

 

Formula

 

Formula

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

 

                                  Formula

                                  Formula

                                    Formula

                                                 Formula

For example, Formula,  but Formula and Formula are not defined.

 

Formula

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

 

Formula

 

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:

                                                        Formula

Examples: Application of definition

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

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

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

 

                                            Formula

Proof: We have 

                                                      Formula

                                                                 Formula

                                                                 Formula

                                                                   Formula

Therefore, we let Formula.

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

                                             Formula

Proof: We have

                                                                   Formula

                                                                 Formula

                                                                 Formula

                                                          Formula

                                                       Formula

Therefore, we let Formula.

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

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

 

                                         Formula

Proof: We have
                                                        Formula

Therefore, we let Formula.

 

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

 

                                        Formula

Prove yourself.

 

Formula

 

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

 

Formula

 

Footnotes

  1. Hypothetical Alert Reader Of Limitless Dedication

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