Question

How do you find `lim (x^2+4)/(x^2-4)` as `x->2^+`?

Answer 1

One method is to evaluate at values closer and closer to 2.

Explanation:

f(2.1) = 20.5
f(2.05) = 40.5
f(2.01) = 200.5
f(2.0001) = 20000.5

So we can see it approaches infinity.

Another method is by graphing. As you can see, the limit approaches infinity.

graph{(x^2+4)/(x^2-4) [-5.74, 12.04, -1.71, 7.18]}

As we can see as x approaches two from the positive direction, the y value seems to go up indefinitely.

Answer 2

Please see below.

Explanation:

As `xrarr2`, the numerator goes to `8` (which is not `0`) and the denominator goes to `0`.

This form `("non-"0)/0` tells us that the function is increasing or decreasing without bound on each side. (The one-sided limits are `+-oo`)

To see which is happening as `xrarr2^+`, we need to determine whether the denominator is going to `0` through positive or negative values.

For `x` a little greater than `2`, we know that `x^2` is a little greater than `4`. Therefore, the denominator is positive.

Something close to `8` divided by a positive number close to `0` (a positive fraction) is a big positive number.

`lim_(xrarr2^+)(x^2+4)/(x^2-4) = oo`

Note it takes a lot longer to explain than it does to do!