Thread: Limits

How can I find the limits of these functions analytically:

1. Limit of (x-3)/(x-2) as x approaches 2 from the right
2. Limit of (x^2)/(x^2-9) as x approaches 3 from the right

I can determine the limit from its graph, but how can I do so without a graphing calculator. This question comes from a chapter about Limits and their Properties. A step by step answer will help a lot. Thanks.

the only thing I can suggest is that you think of it as x = 2+h where h approaches 0 from the right
then you see that the bottom approaches 0 but is always positive
and the top is negative
so -/+ = -
limit -> -infinity

Originally Posted by PMoNEY23

How can I find the limits of these functions analytically:

1. Limit of (x-3)/(x-2) as x approaches 2 from the right
2. Limit of (x^2)/(x^2-9) as x approaches 3 from the right

I can determine the limit from its graph, but how can I do so without a graphing calculator. This question comes from a chapter about Limits and their Properties. A step by step answer will help a lot. Thanks.

I'm not sure what do you mean by "analytically"

But conceptually for #1 would be that as x -> 2+ the numerator would be a number very close to -1 and the denominator would be a very small positive number that is almost=0. Therefore it would be equal to negative infinity.

For number 2, the numerator would be ~9 and the denominator would be a very small positive number that is almost = 0. Therefore it would equal to positive infinity.

I hope this makes sense.

ok so it makes sense but would this be the same for all problems, like for example if the numerator was positive and the denominator negative the limit would approach negative infinity and vice versa?

like in problem 2 where the numerator value is ~9 which is positive, how did u know that the denominator is something positive?

Originally Posted by PMoNEY23

ok so it makes sense but would this be the same for all problems, like for example if the numerator was positive and the denominator negative the limit would approach negative infinity and vice versa?

like in problem 2 where the numerator value is ~9 which is positive, how did u know that the denominator is something positive?

No...the only reason they went to infinity or negative infinity was because the denominator was ~0 and the numerator is a number greater than 0.

Consider this problem
Lim x-> 0 of .00001/x= infinity because as x becomes smaller, the lim becomes larger.

For number 2 I knew the denominator is a infinity small positive number because it the limit was taken from the right side which means (3.00000....1)^2-9= a positive very small number.

Does this make sense?

yes that cleared it up a lot. thanks.