Limit of a Function: De–nitions and Examplesksuweb.kennesaw.edu/~plaval/math4381/func_lim_def_slides.pdfLimit of a Function: Remarks 1 When we say that the limit of a function exists,

Limit of a Function: Definitions and Examples

Philippe B. Laval

KSU

Today

Philippe B. Laval (KSU) Limit of a Function Today 1 / 15

Introduction

Unlike for sequences, there are many possibilities for the limit of a function.In this section, we will investigate the following limits: lim

x→±∞f (x) and

limx→a

f (x). Given a function y = f (x), we are studying how f behaves near

a point a or near infinity. The questions we are trying to answer are:

1 As x is getting closer and closer to a, how is y = f (x) behaving? Is itgetting closer to a number as well? Is it getting arbitrary large (inabsolute value)? Is it not following any pattern?

2 Same question if x is approaching ∞ or −∞.

In more general terms, we are asking the question: if x is following acertain pattern, is f (x) also following a pattern and if yes, which pattern?In order to be able to evaluate lim

x→af (x), f must be defined in a deleted

neighborhood of a that is f must be defined in an interval of the form(a− h, a+ h) for some positive number h, except maybe at x = a.


General Definition of Limits

When we say limx→a

f (x) = L, we mean that f (x) can be made as close as

we want from L simply by taking x close enough to a. Or, in terms ofneighborhoods, we have the following general definition for a limit.

DefinitionWe say that lim

x→af (x) = L or that f (x)→ L as x → a if for every

neighborhood V of L, one can find a deleted neighborhood U of a suchthat x ∈ U =⇒ f (x) ∈ V .

This definition can be adapted to limits at a finite point or at infinity aswell as when the limit is finite or infinite. There are three possibilities forx , we can have x → a, x → −∞ and x →∞. For each case, we can havef (x)→ L, f (x)→ −∞ and f (x)→∞. Hence, we have a total of ninedefinitions. They can all be derived from the above definition simply byremembering that a neighborhood of a finite point a is an interval of theform (a− δ, a+ δ) and a neighborhood of infinity is an interval of theform (w ,∞) for some w ∈ R.


Finite Limit at a Finite Point

We adapt the general definition above in the case limx→a

f (x) = L.


x→af (x) = L or that f (x)→ L as x → a if ∀ε > 0,

∃δ > 0 : 0 < |x − a| < δ =⇒ |f (x)− L| < ε.

|x − a| represents how far x is from a. The above statement says that f (x)can be made arbitrarily close to L simply by taking x close enough to a.

ExampleProve that lim

x→2(x + 5) = 7.

Many of the techniques used for sequences will also be used here.


Infinite Limit at a Finite Point

Recall that a neighborhood of ∞ is an interval of the form (w ,∞).Similarly, an neighborhood of −∞ is an interval of the form (−∞,w).


x→af (x) =∞ or that f (x)→∞ as x → a if ∀M > 0,

∃δ > 0 : 0 < |x − a| < δ =⇒ f (x) > M.


x→af (x) = −∞ or that f (x)→ −∞ as x → a if ∀M < 0,

∃δ > 0 : 0 < |x − a| < δ =⇒ f (x) < M.

Example

Prove that limx→1

1

(x − 1)2=∞.


Limit of a Function: Remarks

1 If we find a δ which works, then every δ′ < δ will also work.Therefore, it is always possible to impose certain conditions on δ suchas saying that we are looking for δ less than a certain number h, thusrestricting our search to an interval of the form (a− h, a+ h). In thisinterval, if we call δ′ the value we found, then δ = min

(h, δ′

).

2 In the last two definition, the vertical line x = a is a verticalasymptote for the graph of y = f (x).

3 In the definition of a limit, it is implied that a is a limit point of D (f )that is for every δ > 0 the interval (a− δ, a+ δ) contains points ofD (f ) other than a. If this is not the case, then for δ small enough,there may not exist any x satisfying 0 < |x − a| < δ. In this case, theconcept of a limit has no meaning.

4 In the definition of a limit, a does not have to be in the domain of f .It only needs to be a limit point of D (f ).

5 In the definition of a limit, δ depends obviously on ε. It may alsodepend on the point a.




(h, δ′

).








(h, δ′

).








(h, δ′

).








(h, δ′

).







1 When we say that the limit of a function exists, we mean that itexists and is finite. When the limit is infinite, it does not exist in thesense that it is not a number. However, we know what the function isdoing, it is approaching ±∞.

2 There are several situations under which a limit will fail to exist.

The function may oscillate boundedly like in f (x) = sin1xas x → 0.

The function may oscillate unboundedly like x sin x as x →∞.The function may grow without bounds like

1x2as x → 0.

There may be a "break" in the graph.







1x2as x → 0.








1x2as x → 0.







The function may oscillate unboundedly like x sin x as x →∞.

The function may grow without bounds like1x2as x → 0.








1x2as x → 0.








1x2as x → 0.



Limits at Infinity

To evaluate limx→∞

f (x), f must be defined for large x , that is we must have

D (f ) ∩ (w ,∞) 6= ∅ for every w ∈ R. For limx→−∞

f (x), we must have

D (f ) ∩ (−∞,w) 6= ∅ for every w ∈ R.


x→∞f (x) = L or that f (x)→ L as x →∞ if ∀ε > 0,

∃w > 0 : x ∈ (w ,∞) ∩ D (f ) =⇒ |f (x)− L| < ε


x→∞f (x) =∞ or that f (x)→∞ as x →∞ if ∀M > 0,

∃w > 0 : x ∈ (w ,∞) ∩ D (f ) =⇒ f (x) > M.


x→∞f (x) = −∞ or that f (x)→ −∞ as x →∞ if ∀M < 0,

∃w > 0 : x ∈ (w ,∞) ∩ D (f ) =⇒ f (x) < M.


Limits at Infinity

|f (x)− L| represents the distance between f (x) and L. The abovestatement simply says that f (x) can be made as close as one wants to L,simply by taking x large enough. Graphically, this simply says that the liney = L is a horizontal asymptote for the graph of y = f (x).To prove that a number f (x) approaches L as x →∞, given ε > 0, onehas to prove that w > 0 can be found so thatx ∈ (w ,∞) ∩ D (f ) =⇒ |f (x)− L| < ε. The approach is very similar tothe one used for sequences.

Example

Prove that limx 7→∞

1x= 0.

Example

Prove that limx→∞

x2 =∞.


Limits at Infinity


x→−∞f (x) = L or that f (x)→ L as x → −∞ if ∀ε > 0,

∃w < 0 : x ∈ (−∞,w) ∩ D (f ) =⇒ |f (x)− L| < ε.


x→−∞f (x) =∞ or that f (x)→∞ as x → −∞ if ∀M > 0,

∃w < 0 : x ∈ (−∞,w) ∩ D (f ) =⇒ f (x) > M.


x→−∞f (x) = −∞ or that f (x)→ −∞ as x → −∞ if

∀M < 0, ∃w < 0 : x ∈ (−∞,w) ∩ D (f ) =⇒ f (x) < M.

Note: In the above definitions, x ∈ (w ,∞) can be replaced byx > w and x ∈ (−∞,w) can be replaced by x < w .


One-Sided Limits

When we say x → a, we realize that x can approach a from two sides. If xapproaches a from the right, that is if x approaches a and is greater thana, we write x → a+. Similarly, if x approaches a from the left, that is if xapproaches a and is less than a, then we write x → a−.We can rewrite the above definition for one sided limits with littlemodifications. We do it for a few of them.


x→a+f (x) = L or that f (x)→ L as x → a+ if ∀ε > 0,

∃δ > 0 : 0 < x − a < δ =⇒ |f (x)− L| < ε


x→a−f (x) = L or that f (x)→ L as x → a− if ∀ε > 0,

∃δ > 0 : 0 < a− x < δ =⇒ |f (x)− L| < ε


One-Sided Limits

Example

Prove that limx→0+

1x=∞.

TheoremThe following two conditions are equivalent

1 limx→a

f (x) = L

2 limx→a+

f (x) = L and limx→a−

f (x) = L

One way to prove that limx→a

f (x) does not exits is to prove that the

two one-sided limits are not the same or that at least one of themdoes not exist.One-sided limits are often used when the definition or behavior of fchanges around the point a at which the limit is being computed as itdoes with piecewise functions.


Examples

ExampleShow that lim

x→3(4x − 5) = 7

Example

Show that limx→2

(x2 + 2

)= 6

Example

Let f (x) =

x if x < 0x2 if 0 < x ≤ 28− x if x > 2

1 Prove that limx→0

f (x) = 0.

2 Prove that limx→2

f (x) doe not exist.


Examples

Example

Find limx→4

√x − 2x − 4 .

The next example illustrates the fact that δ depends not only on ε but alsoon the point at which the limit is being found.

Example

Prove that limx→a

1x=1afor any a ∈ (0,∞).

Example

Let f : R→ R defined by f (x) ={1 if x ∈ Q0 if x /∈ Q . Prove that lim

x→af (x)

does not exist for any a ∈ R.


Exercises

See the problems at the end of my notes on limit of a function: definitionsand examples.


http://ksuweb.kennesaw.edu/~plaval/math4381/func_lim_def.pdf

http://ksuweb.kennesaw.edu/~plaval/math4381/func_lim_def.pdf

Limit of a Function: De–nitions and Examplesksuweb.kennesaw.edu/~plaval/math4381/func_lim_def_slides.pdfLimit of a Function: Remarks 1 When we say that the limit of a function exists,

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