how to solve a limit?

Question

i need to know how to solve a couple problems

x^2-1/x-1 as lim ->1

Answer 1

Using direct substitution you get 0/0 which is indeterminant so you can use l'hospitals rule, which is the easiest way to solve this

take the derivative of the top and bottom and you get
2x/1
now directly substitute and you get
2(1)/1 = 2

Answer 2

x^2-1/x-1 as lim ->1
It looks like the limit as x --> 1 is 0/0.
But this is an ideterminate result. Further investigation is required.
If you factor the numerator you get:
(x-1)(x+1)/(x-1) = x+1
Now you can see that the limit is clearly 2.

Another way is to use L'Hospital's rule:
lim = 2x/1 and as x--> 1 the limit is clearly 2.

L'Hospital' rule says to take the derivative of the numerator over the derivative of the denominator and try the limit again. Repeat until a decision is reached.

Answer 3

Are you asking for the limit as x approaches 1 of (x^2-1)/(x-1)?

You may have already noticed that when x=1, the function becomes 0/0, which is undefined. (This is a good first try because rational functions, such as the function you are asking about, are continuous throughout their domains. If it had been defined at x=1, we would be done.)

Since that attempt failed, let's factor the top using the difference of two squares formula: (x^2-1)/(x-1) = (x+1)(x-1)/(x-1). The two (x-1)'s cancel each other out, leaving us with just the (x+1). And as x approaches 1, x+1 approaches 1+1=2.

Answer 4

sub 1 into the equation wherever you see x, so you get 0/0
that is an indeterminate form which means you will have to work with the equation to find the limit. theres a couple different ways to do it, the method here is to factor the numerator so something will cancel. in this case you get [(x+1)(x-1)]/(x-1)
the (x-1) will cancel and you get the lim->1 of x+1
substitute 1 in and the limit is 2

check it on a graphing calculator if u want

Answer 5

I take it you mean
lim (x->1) of (x^2-1) / (x-1)
= lim (x->1) of [(x-1) (x+1) / (x-1)]
= lim (x->1) of (x+1)
= 1+1
= 2.

Answer 6

I assume you mean

{lim ->1} (x² - 1)/(x - 1)
= {lim ->1} (x - 1)(x + 1)/(x - 1) = x + 1 = 1 + 1 = 2