1. LIMIT OF A FUNCTION AND LIMIT LAWS
1.1 What is Limit
Frequently, when studying a function, we find ourselves interested in the functions behavior near a particular point, but not at itself. This is due to various reasons. One of the reasons could be because when trying to evaluate a function at , it leads to division by zero, which is undefined! Lets have a look at the following example:
Observe the behavior of the following function near.
Notice from graph 1 that there is actually a hole at. Evaluating f at gives:
But is indeterminate, meaning, we cant determine its value. So, is not defined, that is why here is a hole in graph 1.
Right sideLeft sideInstead of evaluating f directly at , lets observe the value of when we approach:
x 0.9999 0.999 0.9 0.5 1 1.0001 1.001 1.01 1.1 1.5
f (x) 1.99990 1.99900 1.90000 1.50000 ? 2.00001 2.00100 2.01000 2.100 2.5
As seen from the table above, the closer x gets to 1, the closer seems to get to 2.Lets generalize the idea illustrated in Example 1:
Limit (informal definition): If the values of can be made as close as we like to L by taking values of x sufficiently close to (but not equal to), then we write
which is read the limit of as x approaches is L.
For instance, in Example 1, we would say that approaches the limit 2 as x approaches 1 and write:
Remember, to find, we should be able to approach as close as we like to. This means: the function f should be defined everywhere near; otherwise, we would say: the limit does not exist. Lets have a look at few more examples.
Graph 2: limit exist because x can approach 2 as close as we like Graph 3: limit does not exist because x cannot approach 2 as close as we like
1.2 One -sided Limits: what is right hand limit & left hand limit?
1.If the value of approaches the number L as x approaches c from the right,
we write(one-sided limit right-hand limit)
2.If the value of approaches the number M as x approaches c from the left,
we write(one-sided limit left-hand limit)
Theorem A function has a limit as x approaches c if and only if it has left-hand and right hand limits and these one-sided limits are equal:
if and only if
If both one-sided limits do not have the same value then does not exist.
Note:The limit of a function as x approaches c does not depend on the value of the function at c [limit does not depend on f(c)]Example 2: Graph 4
At : even though ,
does not exist. The right and left hand limits are not equal.
even though ,
1.3 The Limit Laws
If both exist, and k is a constant, then
If f is the identity function , then for any value of
If f is the constant function , then for any value of
1.4 Computing LimitsThere are various algebraic methods to solve for limits but the first step would always be substitution. If substitution results in
1. A real number, negative infinity (), positive infinity () limit is found
2. Zero over infinity answer to the limit is zero.
3. A real number over zero or infinity over zero Limit does not exist.
Evaluate the following limits.
A) Substitute x with 5:
use the sign graph to determine whether the answer is or both.
does not exist. ( from the right, and from the left)
From the left,
From the right,
Both one-sided limits are the same
However, if substitution results in indeterminate value such as , then we have to convert the function into a suitable form where substitution can give one of the three results mentioned above. The following examples illustrate how to resolve indeterminacy by using algebraic methods.
Example 4 - Resolving indeterminate form of by factorizing:
Substituting x with 2:
Canceling in numerator and denominator, and substituting x with 2:
B) Evaluate Substituting x with 1:
Factorizing intoand to
Canceling in numerator and denominator, and substituting x with 1:
Example 5 - Resolving indeterminate form of by conjugate multiplication:
Substitute t with 0:
indeterminate Multiply with conjugate and simplify:
Cancel in numerator and denominator and substitute x with 0:
Theorem-(The Squeezing Theorem or Sandwich Theorem) Suppose that for all x in some open interval containing c (as shown in graph 5), except possibly at x = c itself. Suppose also that
Graph 5 illustrating squeezing theoremSqueezing theorem helps us to establish several important limit rules such as:
Theorem (limit rules):
Proof:This proof is for the limit as 0+. The case for 0 can be proved in exactly the same manner.Consider the graph to the right. Notice that the area of the triangle OAP is less than the area of the sector OAP which is in turn less than the area of the triangle OAT. Lets estimate each of these areas in turn.
Area of triangle OAP =
Area of sector OAP =
Area of triangle OAT = Using our initial observation, one has
From the left hand side of (1) one has or if . From the right hand side we have or if. In other words,
At this point, if we take the limit as and apply squeezing theorem:
Since the left-hand side and right-hand side has the limit of 1, we can conclude that
1.5 End behavior of function (limits as )
The behavior of a function toward the extremes of its domain is sometimes called its end behavior. In this section, we will use limits to investigate the end behavior of a function as . The symbol for infinity does not represent a real number. We use to describe the behavior of a function when the values in its domain or range outgrow all finite bounds.
Finite limits as (definition):
1. We say that has the limit L as x approaches infinity and write
2. We say that has the limit L as x approaches minus infinity and write
Geometrically, if as, then the graph of eventually gets closer and closer to the line as the graph is traversed in the positive x-direction (see graph 6). And if as, then the graph of eventually gets closer and closer to the line as the graph is traversed in the negative x-direction (see graph 7). In either case, we call line a horizontal asymptote of the graph of f.
Graph 6 Graph 7
horizontal asymptote (definition): A line is a horizontal asymptote of the graph of a function if either
Lets begin by obtaining the limits of some simple functions and then use these as building blocks for finding limits of more complicated functions.
Theorem: let k be a real number.
Theorem: If is a rational number such that is defined for all x, then
The Limit Laws
If both exist, and k is a constant, then
Limits of polynomials as
Note: The end behavior of a polynomial matches the end behavior of its highest degree term.
Limits of rational functions as
There are two methods to find the limit of rational functions as :
Method 1: Use
Example 10: Method 2: Use the end behavior of polynomial at numerator and denominator.
Limits involving radicals
When a polynomial or rational function is below the radical, we can evaluate the limit of the function first and then take the radical.
When radical appears either at numerator or denominator only, it would be useful to manipulate the function to powers of 1/x. This can be achieved in both cases by dividing the numerator and denominator by and using the fact that .
Another simpler method for finding the limit of functions involving radicals is to use the fact that.
, the answer is zero because numerator degree is lower than denominators one.
1.6 Infinite Limits And Vertical Asymptotes
Infinite Limits (definition)
If the values of increase without bound (infinitely) as or ,
If the values of decrease without bound (infinitely) as or ,
If both one-sided limits are , then
If both one-sided limits are , then
Limits as can fail (or does not exist) when
a) the values of increase or decrease without bound.
the graph of the function oscillates indefinitely, the values of does not approach a fixed number, we say that and does not exist.
Vertical asymptote(definition): A line is a vertical asymptote of the graph of a function if either
3. does not exist.
Interior point : A function is continuous at an interior point c of its domain if .
Endpoint: A function is continuous at a left endpoint a or is continuous at a right endpoint b of its domain if
or , respectively.
A function is continuous at an interior point of its domain if and only if it meets the following three conditions:1. exists[ c lies in the domain of f ]2.
exists[ f has a limit as ]3. [ the limit equals the function value ]
If one or more of the conditions are not satisfied, then f is discontinuous at c, and c is a point of discontinuity of f. If f is continuous at (a,b) f is continuous at all points of an open interval (a,b).
If f is continuous on f is continuous everywhere. Types of discontinuity: Removable discontinuity, Infinite discontinuity, Jump discontinuity and oscillating discontinuity.
Continuous FunctionA function is continuous on an interval if and only if it is continuous at every point of the interval.
Theorem: (Properties of Continuous Functions)
If the functions f and g are continuous at , then the following combination are continuous at :1. Sums: 2. Differences: 3. Products: 4. Constant multiples:, for any number k5.
Quotients: ,provided 6. powers:, provided it is defined on an open interval containing c, where r and s are integers.A)PolynomialsAny polynomial is continuous everywhere.
If and are polynomials, then the rational function is continuous wherever it is defined (). The function is discontinuous at the points where the denominator is zero.
is discontinuous at points x = 5 and x = 4.
C)Root FunctionsAny root function is continuous at every number in its domain.
D)Trigonometric FunctionsTrigonometric functions are continuous at every number in their domains.
Theorem:If c is any number in the natural domain of the stated trigonometric function, then
Note:The functions sin x and cos x are continuous everywhere.
E)Composite Functions ()
Theorem:If f is continuous at c and g is continuous at , then the composite is continuous at c
Theorem:If g is continuous at the point b and , then
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