Before proceeding, consider the graph of
shown in
[link] . As
and
the graph of
appears almost linear. Although
is certainly not a linear function, we now investigate why the graph of
seems to be approaching a linear function. First, using long division of polynomials, we can write
Since
as
we conclude that
Therefore, the graph of
approaches the line
as
This line is known as an
oblique asymptote for
(
[link] ).
The graph of the rational function
approaches the oblique asymptote
We can summarize the results of
[link] to make the following conclusion regarding end behavior for rational functions. Consider a rational function
where
If the degree of the numerator is the same as the degree of the denominator
then
has a horizontal asymptote of
as
If the degree of the numerator is less than the degree of the denominator
then
has a horizontal asymptote of
as
If the degree of the numerator is greater than the degree of the denominator
then
does not have a horizontal asymptote. The limits at infinity are either positive or negative infinity, depending on the signs of the leading terms. In addition, using long division, the function can be rewritten as
where the degree of
is less than the degree of
As a result,
Therefore, the values of
approach zero as
If the degree of
is exactly one more than the degree of
the function
is a linear function. In this case, we call
an oblique asymptote.
Now let’s consider the end behavior for functions involving a radical.
Determining end behavior for a function involving a radical
Find the limits as
and
for
and describe the end behavior of
Let’s use the same strategy as we did for rational functions: divide the numerator and denominator by a power of
To determine the appropriate power of
consider the expression
in the denominator. Since
for large values of
in effect
appears just to the first power in the denominator. Therefore, we divide the numerator and denominator by
Then, using the fact that
for
for
and
for all
we calculate the limits as follows:
Therefore,
approaches the horizontal asymptote
as
and the horizontal asymptote
as
as shown in the following graph.
This function has two horizontal asymptotes and it crosses one of the asymptotes.
Determining end behavior for transcendental functions
The six basic trigonometric functions are periodic and do not approach a finite limit as
For example,
oscillates between
(
[link] ). The tangent function
has an infinite number of vertical asymptotes as
therefore, it does not approach a finite limit nor does it approach
as
as shown in
[link] .
Questions & Answers
Three charges q_{1}=+3\mu C, q_{2}=+6\mu C and q_{3}=+8\mu C are located at (2,0)m (0,0)m and (0,3) coordinates respectively. Find the magnitude and direction acted upon q_{2} by the two other charges.Draw the correct graphical illustration of the problem above showing the direction of all forces.
To solve this problem, we need to first find the net force acting on charge q_{2}. The magnitude of the force exerted by q_{1} on q_{2} is given by F=\frac{kq_{1}q_{2}}{r^{2}} where k is the Coulomb constant, q_{1} and q_{2} are the charges of the particles, and r is the distance between them.
Muhammed
What is the direction and net electric force on q_{1}= 5µC located at (0,4)r due to charges q_{2}=7mu located at (0,0)m and q_{3}=3\mu C located at (4,0)m?
Capacitor is a separation of opposite charges using an insulator of very small dimension between them. Capacitor is used for allowing an AC (alternating current) to pass while a DC (direct current) is blocked.
Gautam
A motor travelling at 72km/m on sighting a stop sign applying the breaks such that under constant deaccelerate in the meters of 50 metres what is the magnitude of the accelerate
velocity can be 72 km/h in question. 72 km/h=20 m/s, v^2=2.a.x , 20^2=2.a.50, a=4 m/s^2.
Mehmet
A boat travels due east at a speed of 40meter per seconds across a river flowing due south at 30meter per seconds. what is the resultant speed of the boat
which has a higher temperature, 1cup of boiling water or 1teapot of boiling water which can transfer more heat 1cup of boiling water or 1 teapot of boiling water explain your . answer
I believe temperature being an intensive property does not change for any amount of boiling water whereas heat being an extensive property changes with amount/size of the system.
Someone
Scratch that
Someone
temperature for any amount of water to boil at ntp is 100⁰C (it is a state function and and intensive property) and it depends both will give same amount of heat because the surface available for heat transfer is greater in case of the kettle as well as the heat stored in it but if you talk.....
Someone
about the amount of heat stored in the system then in that case since the mass of water in the kettle is greater so more energy is required to raise the temperature b/c more molecules of water are present in the kettle
pratica A on solution of hydro chloric acid,B is a solution containing 0.5000 mole ofsodium chlorid per dm³,put A in the burret and titrate 20.00 or 25.00cm³ portion of B using melting orange as the indicator. record the deside of your burret tabulate the burret reading and calculate the average volume of acid used?
No. According to Isac Newtons law. this two bodies maybe you and the wall beside you.
Attracting depends on the mass och each body and distance between them.
Dlovan
Are you really asking if two bodies have to be charged to be influenced by Coulombs Law?
Specific heat capacity is a measure of the amount of energy required to raise the temperature of a substance by one degree Celsius (or Kelvin). It is measured in Joules per kilogram per degree Celsius (J/kg°C).
AI-Robot
specific heat capacity is the amount of energy needed to raise the temperature of a substance by one degree Celsius or kelvin
ROKEEB
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