Limits next more questions

[Laven]

1)We know this limit doesn't
$$\lim_{x\rightarrow-\frac{2}{3}}\frac{2}{2+3x}$$ exists
after substituting the value ot that gives us answer infinity.But how about doing derivative at both numerator and denominator that gives us answer to 0.I guess this is not the correct way since I'ven't used the value of x yet,is it?

2)$$\lim_{x\rightarrow\infty}\frac{(2x-10)^6(3x-1)^4}{(2x+1)^10}$$
I got its answer as 81/16 but at book i found the answer is 81/61.Which one is true?Could you please interpret it?I expanded all by bionomial method.Is this true method?If you have next method may i get it please?
[Is there any method to check whether any answer is wrong or right without looking books' answer?]

Seems a lot of questions yet i couldn't solved it.I need your great help.

thanks in advance:p:

 

[tiny-tim]

Welcome to PF!

Hi Laven! Welcome to PF!

(nice LaTeX btw, except that if you have more than one figure after ^, you must put it in curly brackets: ^{10} )

1)… But how about doing derivative at both numerator and denominator that gives us answer to 0.I guess this is not the correct way …

I assume you're thinking of l'Hôpital's rule …

but that only applies to "indeterminate forms" of 0/0, and this is 2/0.

2)$$\lim_{x\rightarrow\infty}\frac{(2x-10)^6(3x-1)^4}{(2x+1)^10}$$
I got its answer as 81/16 but at book i found the answer is 81/61.Which one is true?

Yes, 81/61 is obviously a misprint … 81/16 is correct.

So you're right! ​