A tangent
to a curve is a straight line that touches the curve at a single point but does
not intersect it at that point. For example, in the figure to the right, the
y-axis would not be considered a tangent line because it intersects the curve
at the origin. A secant to a curve is a straight line that intersects
the curve at two or more points.
In the figure to the right, the
tangent line intersects the curve at a single point P but does not intersect
the curve at P. The secant line intersects the curve at points P and Q.
The concept of limits begins with
the tangent line problem. We want to find the equation of the tangent line to
the curve at the point P. To find this equation, we will need the slope of the
tangent line. But how can we find the slope when we only know one point on the
line? The answer is to look at the slope of the secant line. It's slope can be
determined quite easily since there are two known points P and Q. As you slide
the point Q along the curve, towards the point P, the slope of the secant line
will become closer to the slope of the tangent line. Eventually, the point Q
will be so close to P, that the slopes of the tangent and secant lines will be
approximately equal.
A limit of a function is
written as :
We want to find the limit of f(x) as
x approaches a. To do this, we try to make the values of f(x)
close to the limit L, by taking x values that are close to, but
not equal to, a. In short, f(x) approaches L as x
approaches a.
Examples
The
previous example shows that the value a can be approached from both the
left and right sides. Each side has its own limit. For example, as x
approaches a from the right side we have 
and as x approaches a from the left we have 
.
The graph to the right shows an
example of a function with different right and left hand limits at the point x
= 1. As x approaches 1 from the left side, the limit of f(x) approaches 1. As x
approaches 1 from the right side however, the limit of f(x) approaches 4. In
this case, the limit of f(x) as x approaches 1 does not exist, because the left
and right hand limits do not approach the same value. This idea leads to the
following theorem:
For the limit to exist, the left and
right hand limits must approach the same value. In our example, as x approaches
3, the left and right hand limits both approach a value of 4. Since the left
and right hand limits are the same, the limit of f(x) as x approaches 3 exists
and is equal to 4. Even though the actual value of f(3) is equal to 2, the
limit is equal to 4. This gives the following theorem:
Examples
Note: The example above uses basic concepts that are covered in
the sections below.
If a
function is defined on either side of a, but the limit as x
approaches a is infinity or negative infinity, then the function has an
infinite limit. The graph of the function will have a vertical asymptote at a. A curve y=f(x) will have a vertical asymptote
at x = a if any of the following conditions hold: 
Note: When dealing with rational functions, there will always be
a vertical asymptote at values of x that make the denominator equal to 0. This
is due to the fact that the function is undefined at points where the
denominator is 0. However, you must still show that one of the conditions above
holds true to prove that there is a vertical asymptote at that point.
Examples
Calculating limits using graphs and
tables takes a lot of unnecessary time and work. Using the limit laws listed
below, limits can be calculated much more quickly and easily.
From the limit laws above, comes the
property of direct substitution. This property makes it possible to
solve most rational and polynomial functions. The property of direct
substitution states: For any rational or polynomial function f, if a is in the
domain of f then
Often, the method of direct
substitution cannot be used because a is not in the domain of f.
In these cases, it is sometimes possible to factor the function and eliminate
terms so that the function is defined at the point a. For an example of
factoring, see example 6 below.
Examples
4 | Evaluate the limit using limit laws
5 | Evaluate the limit using the property of direct substitution
6 | Evaluate the limit by factoring f(x) and eliminating terms
5 | Evaluate the limit using the property of direct substitution
6 | Evaluate the limit by factoring f(x) and eliminating terms
The squeeze theorem is an important
concept that will be very helpful in upper year calculus courses. The squeeze
theorem states:
In simpler terms, the squeeze
theorem states that if the graph of g is squeezed between the graphs of f and h
when x is near a, and if f and h have the same limit L as x
approaches a, then the limit of g as x approaches a is
also L. The graph to the right illustrates the squeeze theorem.
Note: The function g does not have to be completely contained
between f and h. It must only be contained between f and h while x is
near a. The graph illustrates where the functions f and g cross each
other.
Examples
A function f is continuous at
a number a if:
Note: When proving that a function is continuous, you may only
show that the limit of f(x) as x approaches a is equal to
f(a). This property implies that f(a) is defined and that the
limit exists.
A function is continuous on an
interval if it is continuous at every number that falls within that
interval. Continuous functions have the following properties for simple
operations. If functions f and g are continuous at a, and c is a
constant, then the following functions will also be continuous at a:
A function is discontinuous
at a if it is defined near a but not continuous at a. To
prove that a function is discontinous, we must show which of the requirements
of continuous functions that it fails to hold for. There are several different
types of discontinuity, which are listed below.
Removable discontinuity: A function has a removable discontinuity at a if
the limit as x approaches a exists, but either f(a) is
different from the limit or f(a) does not exist. It is called removable
discontuniuity because the discontinuity can be removed by redefining the
function so that it is continuous at a. In example #6 above, the
function has a removable discontinuity at x = 3 because if the function is
redefined so that f(3) = -4/7, it will be continuous at x = 3.
Infinite discontinuity: A function has an infinite discontinuity at a if
the limit as x approaches a is infinite. In example #3 above, the
function has an infinite discontinuity at every point a = k*pi, since each
point has an infinite limit.
Jump discontinuity: A function has a jump discontinuity at a if the
left- and right-hand limits as x approaches a exist, but are
different. It is called jump discontinuity because the function jumps from the
left-hand limit to the right-hand limit at each point. In example #2 above, the
function has a jump discontinuity at x = 0, since the right and left hand limits
approach different values.
Note: Polynomial functions are continuous everywhere. Rational,
root and trigonometric functions are continuous at every number in their
domain. In other words, they are continuous wherever they are defined.
Examples
8 | Determine where the function is continuous
9 | Explain why the function is discontinuous at each point
9 | Explain why the function is discontinuous at each point
The intermediate value theorem
states: If f is a continuous function on the closed interval [a, b]
and N is a number between f(a) and f(b), where f(a)
does not equal f(b), then there exists a number c in interval (a,
b) such that f(c) = N.
The
intermediate value theorem simply says that if a function is continuous on a
closed interval [a, b], and f(a) is different from f(b),
then every value on the y-axis between f(a) and f(b) has a
corresponding value on the x-axis between a and b.
Note: As the graph to the right illustrates, there can be more
than one value c in (a, b) for a number N betwen f(a)
and f(b). The graph shows that there exist 3 numbers, c0,
c1 and c2, such that f(c0)=f(c1)=f(c2)=N.
The intermediate value theorem is
helpful when proving that a root of a function exists in a certain interval.
Examples
As
explained at the beginning of this tutorial, a tangent to a curve is a line
that touches the curve at a single point, P(a,f(a)). The tangent
line T is the line through the point P with the slope :
given that this limit exists. The
graph to the right illustrates how the slope of the tangent line is derived.
The slope of the secant line PQ is given by f(x)-f(a)/x-a.
As x approaches a, the slope of PQ becomes closer to the slope of
the tangent line T. If we take the limit of the slope of the secant line
as x approaches a, it will be equal to the slope of the tangent
line T.
The slope of the tangent line
becomes much easier to calculate if we consider the following conditions. If we
let the distance between x and a be h, so that x=a+h,
and substitute that equality for x in the slope formula, we get:
Note: Either of the limit formulas above can be used to find the
slope. You will obtain the same answer using either formula.
These formulas have many practical
applications. They can be used to find the instantaneous rates of change of
variables. For example, if we use the formula above, the instantaneous velocity
at time t=a is equal to the limit of f(a+h)-f(a)/h
as h approaches 0.
Examples
11 | Find the equation of the tangent line to the
curve at the point P
12 | Find the instantaneous rate of change
12 | Find the instantaneous rate of change
For more practice with the concepts
covered in the limits tutorial, visit the Limit Problems page at the link
below. The solutions to the problems will be posted after the limits chapter is
covered in your calculus course.
To test your knowledge of limits, try taking the general limits test on the iLrn website or the advanced limits test at the link below.
To test your knowledge of limits, try taking the general limits test on the iLrn website or the advanced limits test at the link below.
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