Mastering Limit Laws to Correct Answers | Checking Limits"

[phrox]

First of all, I have to use all the limit laws I can to get these answers correct as the prof said.

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

I factored out x^2, put the x^2 in front of the limit(constant multiple law), plugged in 1/3 into the x's, then multiplied everything together and got 1/3 for my answer. Is the only limit law that can be used the constant multiple law?

2)
lim (3x^(2/3) - 16x^-1
x->8

I don't even see any laws I can use in this, so I just plugged in 8, did the powers and third root of 64, etc etc. and got my final answer to be 10. There must've been a law I could've used, is there?

3)
lim (sqrt(w+2)+1) / (sqrt(w-3)-1)
w->7

I think I'm over-complicating this one, can I just use the quotient law and just plug the w in and solve by dividing top by bottom? This is how I tried to do 3:
multiplied by the top conjugate, so I multiplied top and bottom by sqrt(w+2)-1 and everything just went too big and complicated. Any help?

Thanks so much!

 

[Evgeny.Makarov]

First of all, I have to use all the limit laws I can to get these answers correct as the prof said.

To say whether your answers, or, rather, solutions, are correct, we need a complete list of limit laws you are allowed to use. For example, there is a law saying that if a function is continuous at a point, then its limit at this point coincides with its value. Using this law, all your problems are trivial, so I assume it is not allowed.

The numerical answers for 1) and 2) are correct.

I factored out x^2, put the x^2 in front of the limit(constant multiple law)

$x^2$ is most definitely not a constant because $x$ is tending to 1/3.

 

[phrox]

The laws I can use are:
Sum,Power,Roots,Constant Multiple, Product,Quotient. That's all I have in my notes.

So have I done #1 wrong since I can't use the constant multiple law?

 

[Evgeny.Makarov]

The laws I can use are:
Sum,Power,Roots,Constant Multiple, Product,Quotient.

Strictly speaking, this is not enough. It does not even allow finding $\lim_{x\to3}x$. However, if you have the idenity function law (my name) that $\lim_{x\to a}x=a$, then you can do the problems just by parsing the expression and applying the correspoding law. If the top-level operation (the one you do last when you compute the value of the expression with a known $x$) is +, you apply the sum law; if the top-level operation is square root, you apply the root law, etc. The only thing is that you need to make sure that the limits of the smaller expressions exist and the operation with them makes sense (e.g., the limit of the denominator is not zero). You find this out when you parse the expression to the bottom.

So have I done #1 wrong since I can't use the constant multiple law?

You just use the product law instead. Also, there is no need to change the expression of which you take the limit (like factoring out $x^2$).

 

[phrox]

Oh okay, so for the 3rd question, is it correct just by using the quotient law?

 

[Evgeny.Makarov]

so for the 3rd question, is it correct just by using the quotient law?

Yes, since the limit of the denominator is not zero.