Algebra vs. Calculus :}

1. What is the DIFFERENCE between finding the limit of a function at x=c and actually plugging in the number x=c? When are the two cases the SAME?

• When finding the limit of a function at x=c, you're finding what y value f(x) gets closer to as it approaches c. There doesn't actually have to be a point at x=c. There can be a hole at x=c and the limit can still exist, as long as f(x) passes through only one point at x=c.
• In turn, when plugging in the number x=c, there cannot be a hole at that point. You are finding the exact value of y when x equals a constant. Even though there cannot be a hole, the function doesn't have to actually pass through the point. There could be 2 open dots, one coming from the left and approaching a value different from f(c) and one coming from the right and approaching an entire new value from the other two. As long as there is a closed dot at x=c, f(c) does exist.
• They are the same when there is continuity at x=c. If the lim f(x) x->c = f(c), then you will get the same value for both and therefore f(x) does pass through the exact point at x=c.

2.What are the SIMILARITIES between finding the derivative and finding the slope of a line? What are the DIFFERENCES between the two?

• The similiarities between finding the derivative and finding the slope of a line is that you use change of y/change of x to find the slope, even though the specific formula for finding them are different.
• For finding the derivative, limits are involved. One of the most commonly used formulas is limf(x)h->0 = f(a+h)-f(a)/h. To find the derivative, you are bringing one point closer and closer to a main point and a tangent line is formed. This is done by making h approach 0, which closes the gap between the 2 points since h is the distance between the first point and the main point. With a derivative, you are finding the slope of a tangent line on a curve at a specific point (depending on where they want you to find the slope). There can be many different slopes on the curve as well.
• When finding the slope of a line, you are finding only the slope of that line. It is a specific slope (unlike the many slopes you can find when finding the derivative of a curve).

ex. 2x+1. The only slope on this line is 2.

x^2. This parabola has many slopes with different ones at each point on the parabola where the tangent lines are formed.

I've Reached My Limit! >=/

What do you still not understand about limits? Choose 3 problems or types of problems anywhere from chapter 2 that were the most difficult and that you would like to get more help on. OR you can simply explain 3 ideas / concepts that still elude you.

1. I still find it sort of confusing at times to find the limit of something as it approaches infinity and negative infinity. It is not difficult with a graph in front of me, or if it is a simple equation. But when the equation gets really difficult or confusing, I do understand that you have to find the end behavior model and go from there. Sometimes, I just get confused and I would like more practice on this in order to understand it even better.
2. When it comes to finding vertical asymptotes, I understand it most of the time. What I do have trouble with sometimes is finding the horizontal asmyptotes. I would understand this concept and forget it continuously, and I think that the only way to engrave this in my head is by practicing it more. On friday by 4th period after various attempts to understand horizontal asymptotes, I recalled that the limit of f(x) as x approaches + and - infinity=c, then c is the horizontal asymptote. I also remember that if you find the end behavior model, you can just imagine the graph in your head and see where the horizontal asymptotes are, if any.
3. At times I get confused on how to "Describe the behavior of f(x) to the left and right of each vertical asymptote". I know how to do this, but when I cannot imagine the graph in my head, I don't know whether it approaches negative or positive infinity from the left or from the right of the vertical asymptote. It is difficult to determine this on my own, like on:

f(x)={x^3-4x, x<1

{x^2-2x-2, >or= to 1

This problem also greatly affects me with piecewise equations when it asks to find the limit of f(x) as x approaches +or-c from the +or- side of a equation that has a difficult graph to imagine, and I have to say whether it is negative or positive infinity. Sometimes end behavior isnt enough for me to figure it out, so if it is, I would like to have that concept explained to me better.