Could somebody explain to me how to find the limit of a function?

Question

For example, with the limit (2-x)/((x-1)^2)...with x approaching 1, what would be the limit?

There is no summer math tutor available for summer courses at my college to ask these questions...I'm not just looking for homework answers. I appreciate any help, thanks!

Answer 1

You are dealing with a fraction here.

If the numerator becomes zero and the denominator is not zero, the limit is zero.
If the denominator becomes zero and the numerator is not zero, the limit is +/- infinity.

If both numerator and denominator become zero, it is more complicated. If top and bottom are polynomial functions, the one with the highest degree is "stronger": if it is located in the numerator, the limit is zero; if in the denominator, the limit is infinity. If they have equal degree, divide the leading coefficients.

Examples:

a) lim{x -> 1} (2-x) / ((x-1)^2) would give 1 / 0 = infinity

b) lim{x -> 1} (1-x) / ((x-1)^2) would give 0 / 0.
The top function has degree 1, the bottom degree 2, so the bottom function has the strongest zero --> infinity.

c) lim{x -> 1} (4x^3 - 4x) / (x^3 + 2x^2 - 5x + 2) would give 0 0.
The functions both have degree 3, so divide the leading coefficients: 4/1 = 4.

Answer 2

In this case you expand the function and apply the limit
It looks like the limit is 0 by my memory. You have to expand (x-1)^-2 apply the limit

Answer 3

The easiest way in this case is substitution...if you plug in x=1, you get 1/0.

If you get any number divided by zero, then the limit does not exist.

If you substitute the number in, and get something like 2/3, then that is the limit.

If you substitute the number in, and get 0/0, then factor the numerator and denominator, simplify, and then plug in the number to find the limit.

Answer 4

if you put x=1, in given function then it will reduce to 1/0 . therefore limit of this function does not exist.

Answer 5

The expression is not complete.