Algebra Vs. Calculus

1. Depending on the situation, there may or may not be a difference between the limit of a function at x=c and plugging in the number x=c. When finding the limit at x=c, one is looking for the number the function outputs as the input approaches c. There may be an hole at x=c, but if the function approaches that point from both sides, it is indeed the limit. When searching for f(c), one is looking for the output at the x value c. The output does not necessarily have to be the same as the limit of x=c. However, if the limit at x=c equals f(c), then the function is continuous.

Only when the function is condtinuous does the limit at x=c equal f(c).


2. When finding the slope of a line, one is finding the (change in y)/(change in x) of the entire line, not just between two pints. To find the slope of a line, one uses the formula (change in y)/(change in x). (y2-y1)/(x2-x1), another variation of this formula, is sometimes used. Finding the slope of a line typically does not require much more than counting rise over run or plugging in numbers into a simple formula.

To find the derivative of a line, one has the choice of using various different formulas, three of which involve limits, and one which does not. The typical, most basic formula is limit as h approaches 0 f(h+a)/h. When finding the derivative, one is basically finding the slope of a line at one specific point (this is why h is approaching 0).

Finding the slope and finding the derivative of given lines are similar in the way that both of them deal with finding the change in x and y.

I've Reached My Limit !!

Limits seem to perplex me a bit. I understand the basic concept of limits, however, i do not understand certain small details.

1. I am confused on how to find the limit of a function as it approaches a vertical asymptote. The way i solve such problems is by plugging in numbers in order to get an idea of how the graph looks like. Then, i can easily see the right- and left- hand limits of a certain point. I do not think this is how to properly solve it, for it is too time consuming. I was wondering if anyone knew how to solve such problems quickly and easily ??
   
2. I am a little unclear on how to find the limit of a sine or cosine function. I am not sure whether the limit is simply positive and negative one, or just positive one or just negative one.
3. I do not completely understand how to find out if a discontinuity is removable or non-removable. An example of such a problem would be # 25 from homework g1. I also do not know how to find values of x for which a function is discontinuous. An example of such a problem would be part e on question 18 of the test on chapters 2.1-2.3.
   
4. I do not understand how to solve #3 from the test on chapters 2.1-2.3.

Thanks for considering my problems ! :]