We find limits using numerical information.
Numerical Limits
In what follows, functions will be presented using formulas. We will determine the limit of a function by making an appropriate table of values.
Now, to determine the limit numerically, we will plug these four values of into the function and look for a pattern in the outputs.
Here is the analysis:
If then .
If then .
If then .
If then .
We can summarize this information in a table:
The outputs appear to be approaching the number 11, leading us to conclude that
In this particular example, we could have arrived at the answer by simply plugging the number into the function since .
In general, when plugging in the value that is approaching yields a number, that number is usually the correct answer.
We now turn our attention to examples where the shortcut of “plugging in” does not work.
Since we know that is approaching the number from the right side, so . To represent this numerically, consider the following values for : . These numbers are each greater than 1 and they are getting closer to 1.
To determine the limit numerically, we will plug these four values of into the function and look for a pattern in the outputs.
Here is the analysis:
If then .
If then .
If then .
If then .
If then .
We can summarize this information in a table:
The outputs appear to be approaching the number 1.5, leading us to conclude that
Now determine the limit:
The values of the function are are all equal to , and so we can conclude that It
should be noticed that the shortcut of plugging in yields the meaningless
indeterminate form .
Now determine the limit:
The next example comes from solving a tangent line problem.
Based on the numerical information provided in the table above, we conclude that
Observe that plugging into the function yields the indeterminate form , which is not the answer we sought.
Now determine the limit:
Based on the example above and the result of this problem, determine the two-sided limit:
The limit is (DNE is a possibility)
Now determine the limit:
First, note that the shortcut of plugging into the function yields the indeterminate
form .
Hence, to compute this one-sided limit, we consider the following values for : and .
Plugging these values into the function, we generate the following table of
values:
Based on this numerical evidence, it would be reasonable to guess that
Now determine the limit:
Based on the example above and the result of this problem, determine the two-sided limit:
The limit is (DNE is a possibility)
Now determine the limit:
The numerical evidence suggests that as approaches from the left, the values of are decreasing without bound. We conclude that
This result has geometric significance. It means that the line is a vertical asymptote for the graph of the function
Now determine the limit (type infinity for and -infinity for ):
The numerical evidence suggests that as approaches from the left, the values of are increasing without bound. Therefore, we are led to conclude that
This result has geometric significance. It means that the graph of the function has a vertical asymptote at
Now determine the limit (type infinity for and -infinity for ):
In the following examples, we discuss limits as the input approaches either or . If then is increasing without bound and we can use very large numbers for in our table. Similarly, for .
Since , we will use powers of ten to generate large values of when constructing our table:
The numerical evidence suggests that as approaches , that is, as increases without bound, the values of are approaching the number . Hence,
This result has geometric significance. It means that the line is a horizontal asymptote for the graph of the function
Since , we will use negative powers of ten when constructing our table:
The numerical evidence suggests that as approaches , that is, as decreases without bound, the values of are approaching the decimal which we recognize as the fraction . Hence,
This result has geometric significance. It means that the line is a horizontal asymptote for the graph of the function
Compound Interest and the Number
and we will try to find
To do this, let’s construct a table of values with large numbers for :
What we can see from this table is that even with the number of compounding periods being as large as 1,000,000 per year, the principle of $1 will not even grow to $3 by the end of the year. So there appears to be a limit to the effect of increasing the number of compounding periods on the amount of money generated by compound interest at a rate of 100%. This limit is the famous number that we see in the exponential function and as the base of the natural logarithm, .
In conclusion, we have discovered the famous limit
This limit is one way to define the number .
This number was coined by the renowned Swiss mathematician Leonhard Euler in the eighteenth century. He used the symbol to stand for ”exponential” and to fifteen decimal places, the (irrational) number is:
Now determine the limit (the answer is a well known number, denoted by a single letter):
Rectilinear Motion and Instantaneous Velocity
Rectilinear motion is motion along a straight line. We will consider an object to be in motion along a number line to keep track of its location. We let the function denote the position of the object at time . Then the displacement of the object over a time interval, is given by . If we divide the displacement by the duration of the time interval, we get the average velocity of the object over that time interval:
The average velocity is ft/sec.
Our goal is to determine the instantaneous velocity of the object at a given time. To do this, we will consider time intervals of shorter and shorter duration.
The average velocity of the object over the time interval is given by since feet. To obtain the instantaneous velocity, we will look for a pattern in the average velocity as the time interval gets shorter and shorter: It appears that as the average velocities are approaching the value ft/sec, as the time intervals get shorter and shorter (negative velocity just means the object is falling). Hence we conclude that the instantaneous velocity at time seconds is ft/sec. Moreover, in terms of limits, the instantaneous velocity is a limit of average velocities: Technically, in this example we only considered the left hand limit, . In the next problem, we will verify that the right hand limit gives the same value.
Now determine the instantaneous velocity as a limit of the average velocities: