Evaluating Limits: Understanding and Solving Common Problems | Explained

[Saru]

Evaluate the following limits

a) lim x->1 ((x^(1/3) -1) / (x^(1/2) - 1))

b) lim x->0 (( l 2x-1 l - l 2x+1 l ) /x)
(FYI : there's modulus at 2x-1 and 2x+1 )

c) lim x->0 ((sin(a+2x)-2sin(a+x)+sin a) / x^2 )

Thankz
Please explain in details

 

[lanedance]

Hi Saru, welcome to PF

the idea is to attempt the problem and someone will try and help you through - so you have any working or ideas?

 

[CompuChip]

Also, please provide some relevant information. For example, do you need to show the limits from the definition? Are you allowed to use L'Hopital's rule?

 

[Saru]

any methods or working will do.. and I've totally no idea how to do it..
Plus my assignment is due this Wed..

 

[CompuChip]

It's good that you have asked for help this early then, it will give you two days to finish it.

So a good first try is always to check if you can't just plug in the numbers.
For example
$$\lim_{x \to 0} \frac{x^2 + 7x - 8}{\sqrt{x^3} - 3 \sin(x) + 7 \cos(x) + 1}$$
looks terrible, but plugging in x = 0 shows that the limit is -1.

Did you try that already?

 

[n!kofeyn]

any methods or working will do.. and I've totally no idea how to do it..
Plus my assignment is due this Wed..

You need to be sure you know what l'Hopital's rule is before you use it, as well as being absolutely sure that it is valid on your assignment. If you are sure you can use it, then both (a) and (b) can be done using l'Hopital's rule.

 

[Saru]

just found out that I'm not suppose to use Lhopital rule to solve it..
any other methods??

 

[lanedance]

do you have any ideas or attempts?

here's some to get you started, but you need to show you're trying

for a) L'Hopital would work quite easily, though you can't use it...

for b) try writing out the modulus as a normal function as it behaves in a region around x = 0. This shouldn;t be an issue as the modulus only has a kink where the argument changes from poistive to negative

with arguments 2x-1 and 2x+1, this will only be around the points x=1/2 and x= -1/2

for c) as a start, i would try expanding the expressions in terms of angle sum and double angle formula, to see if that helps