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Discussion
Using the draggable points below, estimate the value f(5).
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On the graph, the closest distance we could check was 0.05. What values would you expect to see for f(left) and f(right) if we could see a distance of 0.001?
Part (a)
At x=5, the graph shows the point (5,f(5)) on the curve. By inspecting the graph, the y-coordinate at this point appears must be between 2.31 and 2.47.
To estimate f(5), take the average (midpoint) of these two values:
So, a reasonable estimate is 2.39.
Part (b)
As the horizontal separation shrinks, the two y‐values on either side of x=5 get ever closer to f(5), which we estimated in part (a) as 2.39. So at a distance of 0.001 you would expect
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We just explored the concept of a limit. The basic idea of a limit is that as x approaches a given value a, f(x) approaches a value too.
Basic concept of a limit
The limit x→alimf(x) is the value f(x) approaches as x approaches a.
Types of limits:
Trivial limit: a limit that is simply equivalent to the value of the function evaluated at a.
Rational limit: a limit of a rational function as x approaches a value at which the function is undefined.
Infinite limit: a limit as ∣x∣ gets arbitrarily large. If a function grows without bound as ∣x∣ gets larger, the limit evaluates to ∞ (or −∞). If the function has an asymptote y=k, the limit is k. Infinite limits are undefined for oscillating functions since they do not approach a single value.
Checkpoint
Calculate x→−∞limex.
Select the correct option
Each type of limit may be interpreted graphically:
Trivial limits are simply the value f(a).
Rational limits will appear as vertical asymptotes or holes on a graph.
Infinite limits may be evaluated as a value if there is a horizontal asymptote or ±∞ if a curve increases or decreases without bound.
Limit from a graph
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Discussion
Values of a function f are shown at values of x surrounding 2.
From the table, find x→2limf(x).
As the table shows, whenever x is closer to 2 (whether just below or just above), the corresponding values f(x) get closer to 0. For example:
In each case, as x moves nearer to 2, f(x) moves nearer to 0. Hence
Limit from a table
Given a table of values:
Discussion
One key use of limits is in defining the derivative of a function f at x=c, which is given by
Aside from the limit, does this expression look like anything we have seen before? Recall that (c,f(c)) and (x,f(x)) are points on the graph of f.
The expression inside the limit is just the “rise over run” formula for the slope of the line through (x,f(x)) and (c,f(c)). If we set
then
which is exactly the familiar formula for the gradient of the line passing through those two points.
Discussion
We learned how to define the limit of a function at a point with
but the following equation defines the derivative for the entire function f:
Show that the new equation for the derivative is also a slope between two infinitesimally close points.
Using the new definition of the derivative, find f′(x) given that f(x)=x2.
Part (a)
The definition
is another way to express the slope between two points on the graph of f that are very close together.
Recall that the slope between two points (x1,y1) and (x2,y2) is given by
If we let x1=x and x2=x+h, then y1=f(x) and y2=f(x+h). Substituting into the slope formula gives
So, the expression inside the limit in the definition of the derivative is just the slope between the points (x,f(x)) and (x+h,f(x+h)).
Taking the limit as h→0 means we are finding the slope between two points that are infinitesimally close together. Therefore, the definition
still represents the slope between two infinitesimally close points on the curve.
Now, it is reasonable to conclude that even though the two equation for the derivative look disimilar, they are indeed equivalent since they both give the slope of a curve "at a point."
Part (b)
Using the definition
with f(x)=x2, write
Expand in the numerator:
So
Answer: 2x
Limit definition of derivative
The derivative of f(x) is denoted f′(x) and is given by
The derivative returns the slope "at a point" (or instantaneous slope) on the curve f.
Discussion
Using the limit definition of a derivative, show that the derivative of any line is its slope.
Explain why it makes sense that the derivative of a line is constant.
Part (a)
Let f(x)=mx+b. Then by definition
In slope-intercept form m gives the slope of the line, so the derivative of a line with slope m is m, as desired.
Part (b)
The derivative measures the instantaneous rate of change (or “steepness”) of a function at each point. For a line f(x)=mx+b, the graph of f is a straight line of slope m. Thus, the instantaneous slope at any point on the curve must be m.
In other words, since the rate of change of a line is constant, its derivative is constant.
Suppose we want to solve a limit of the form x→alimg(x)f(x), but x→alimf(x)=0=x→alimg(x). Our limit evaluates as 00. In such a case, if f′(a) and g′(a) exist, then we must have f(a)=0=g(a). We can still solve the limit using the derivative:
Therefore, whenever we take a limit whose numerator and denominator both evaluate to 0, we can equivalently take the limit of the derivatives of the numerator over the derivative of the denominator. This is called L'Hôpital's rule.
Discussion
Using L'Hôpital's rule, show that we can evaluate x→alimg(x)f(x) if x→alimf(x)=∞=x→alimg(x) using derivatives.
Reduce ∞/∞ to 0/0 by taking reciprocals. Define
so that as x→a, h(x)→0 and k(x)→0. Then
and L’Hôpital’s rule for the 0/0 case gives
This is all that is necessary to correctly answer the question above. However, by expanding x→alimh′(x)k′(x), it is possible to show that L'Hôpital's rule applies the same for 00 and ∞∞ limits!
L'Hôpital's rule
If g(a)f(a)=00 or ∞∞, then:
Examples
Since e2x→∞ as x→∞:
Similarly, e0−0−1=0 and 02=0:
Sometimes it's easier to factor out common terms than to use l'Hôpital's rule:
So
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