L'Hopital's Rule- Sean Dandeneau

L’Hopital’s Rule

We recently learned how to use L’Hopital’s rule when we must do so. This situation appears whenever we try to evaluate limits that produce indeterminate forms when we try direct substitution.

To use L’Hopital’s rule, you must find the derivatives of the numerator and the denominator. Once this is found, you plug in c for  x and solve for the limit as x->c.

Lim of f(x)/g(x) as x->c = lim of f’(x)/g’(x) as x->c = lim of f’’(x)/g’’(x) as x->c

Lim 1-cos(x)/x as x->0

= 0/0

    So we must use L’Hopital’s Rule…

Lim 0-(-sin(x))/1 as x->0

= lim sin(x) as x->0

= sin(0)

= 0

When doing L’Hopital’s Rule, we must keep in mind that all derivative rules apply. We can find as many derivatives as we need to in order to solve for the limit at c.

Another example:

f(x)=lnx^5/x

Lim x->infinity lnx^5/x

    = lim x->infinity 5lnx/x

    = lim x->infinity (5/x)/1

    = lim x-> infinity 5/x

    = lim x-> infinity 0/1

    = 0

Continuity and Discontinuity

Over the weekend, our homework was to complete a survey evaluating limits through the topic of roads and bridges. On Monday, we began our discussion on continuity by thinking in these roads and bridges terms:

Continuity- a function f(x) is continuous at x=a if

1. the roads meet
2. there is a bridge
3. the roads meet at the bridge

After conceptually understanding what continuity is, we decided to translate this definition into calculus lingo.

Continuity- a function f(x) is continuous at x=a if

1. the limit of x as it approaches a exists
2. f(a) exists
3. the limit of x as it approaches a is equal to f(a)

In other words

• If you can draw it without lifting your pencil, it’s continuous.
• f(x) is continuous at x=a if
• the limit from the left equals the limit from the right (or the roads come to the same spot)
• a y-value exists at x=a (bridge exists; there isn’t a hole or asymptote there)
• the y-value, the limit from the left, and the limit from the right are all the same number

Discontinuity

• A function has a point of discontinuity at x=a if it fails one or more of the four criteria listed below

The Four Types of Discontinuities and How to Find Them:

1. Removable
1. Piecewise function
2. Hole (a factor cancels)
2. Jumps
1. Absolute value division function
2. Greatest integer function
3. Piecewise function
3. Infinite
1. y=tan(x) (infinite discontinuities at odd integers of pi/2)
2. Reciprocal (vertical asymptote- cannot cancel an x in the denominator)
4. Restricted
1. Square root (continuous for all x-values in its domain, but not outside its domain)
2. Logarithms (only continuous for x>0, unless it is transformed)

The worksheet for discontinuity:

Based on the definition of continuity and the four types of discontinuity, we were able to solve the following problems in class:

To find a value that makes a function continuous, plug in the the boundary (in this case, the boundary is 1) into the x of each part of the piecewise function and solve. Then, set up the answers from each part of the piecewise function and save for a, as shown above. 

Here are answers to the homework problems!

Differentiability

Differentiability: f(x) is not differentiable at x=a (f’(x) does not exist) when f’(x) is not continuous at x=a.

For example:

The Four Ways a Function can Fail to be Differentiable at x=a:

1. A corner
1. Different one sided derivatives causes the graph to jump (ex. absolute value graph)
2. A vertical tangent
1. One-sided derivatives go toward infinity (ex. y=x^⅓)
3. A cusp
1. One-sided derivatives go toward opposite infinities (one toward positive infinity and the other toward negative infinity)
4. Any type of discontinuity
1. Any of the four types of discontinuity explained in the previous section!

The worksheet for differentiability:

Remember:

• Continuity does not imply differentiability, but
• Differentiability does imply continuity.

In other words:

• f’(c)=lim x as it approaches c [f(x)-f(c)]/(x-c) must exist if f is differentiable at x=c

Based on these notes of differentiability, we were able to complete the following in class problems:

To make it differentiable, you find the derivative of the top part of the piecewise function and the derivative of the bottom of the piecewise function, set them equal to one another, and then solve. In this case, you also had to make it continuous, which is explained in a problem above. 

Here are the homework answers!

Reminder

Don’t forget to study for the assessment on tuesday!

Blog for November 4

Overview:
This week we began learning about limits. We had seen limits before, at the beginning of the derivatives unit. The limit definition of the derivative of a function at a point helped us find the slope of the tangent line at x=c. The limit definition of the derivative of a whole function was used to find the slope of a secant line between any point on f(x) and another point that is h units away from it.
As we move away from the topic of derivatives, we focused on limits as x approaches any point in a function.

Graphical Approach
There are three main approaches to limits. We began with the graphical approach. This entails using graphs to find the limit at a point.


We were asked to find the limits from the left and right of each point pictured, as well as the actual limit for each value.


We learned that the limit does not exist if the limits approaching from the left and right sides are different from each other. The limit also does not exist if either of these values is infinity or negative infinity. In other words...


We then did some practice finding limits using graphs.



Numerical Approach
The next approach is the numerical approach. For this approach, we use a table to determine the limits approaching a point from the left and right. As the value gets closer to x, the limit gets closer to what it should be. For example:

In this example, the limits on either side seem to be approaching the same number, .333. This means the limit as x is approaching 2 is .333. If the limits on  either side are not approaching the same number, or they are approaching infinity or negative infinity, the limit does not exist.


Analytical Approach
The last approach is the analytical approach. This is when you use direct substitution to find the limit. You plug in the value, c, into the function, f(x). For normal functions, if the limit as x approaches c exists, it should be f(c).
 
In the last example, the limit ends up being 0/0. This means that there is a hole in the graph at this point.



We were also introduced to the properties of limits in class, which are listed below.


The last topic we covered this week was limits at infinity. This involves plugging infinity into functions as the x-value,  which would result in finding the end behavior of functions. To find the end behavior or a horizontal asymptote, you need to find the limit as x approaches both infinity and negative infinity.
 

Reminder: Monday's class will be a review for our quiz on Thursday.