Evaluate the limit and justify each step by indicating the appropriate limit laws

lim as u approaches -2 square root of (u^4 +3u +6)

Question

Evaluate the limit and justify each step by indicating the appropriate limit laws

lim as u approaches -2 square root of (u^4 +3u +6)

anonymous · Answer

$$\lim_{u \rightarrow -2} \sqrt{u ^{4}+ 3u+6} $$

jim_thompson5910 · Answer

replace u with -2 and evaluate like normal

anonymous · Answer

so is it 4

anonymous · Answer

and what limit laws were used

jim_thompson5910 · Answer

yes the answer is 4


the limit law used was 


$$\Large \lim_{u\to c}(f(u)) = f(c) $$


f(u) in this case is $$\Large f(u) = \sqrt{u^4+3u+6}$$

anonymous · Answer

does that limit law mean that the limit of a constant is the constant of its limit?

anonymous · Answer

and can u help me w/ anotehr limit problem

jim_thompson5910 · Answer

no, that law is 

$$\Large \lim_{u\to c}(k) = k$$

anonymous · Answer

oh so what does that one taht u wrote mean verbally

anonymous · Answer

limu→c(f(u))=f(c)

jim_thompson5910 · Answer

It's known as the direct substitution property

jim_thompson5910 · Answer

your book/teacher may have another name for it

anonymous · Answer

oh okay...can u help me with anotehr limit problem

jim_thompson5910 · Answer

sure

anonymous · Answer

so in this limit i evaluated it and i got 0 over 0 which is indeterminate. my teacher always says to somehow simplify teh problem if it is indeterminate

$$\lim_{x \rightarrow -4} \sqrt{x ^{2}+9} - 5 / (x+4)$$

jim_thompson5910 · Answer

are you familiar with L'Hospital ?

anonymous · Answer

no..what is taht

jim_thompson5910 · Answer

ok nvm that option then

jim_thompson5910 · Answer

what we need to do here is rationalize the numerator, how do we go about doing that?

anonymous · Answer

arent we supposed to multiply the top and bottom by teh conjugate (x-4)

jim_thompson5910 · Answer

that's if we wanted to rationalize the denominator

jim_thompson5910 · Answer

instead, we multiply top and bottom by $$\Large \sqrt{x^2+9}+5$$

jim_thompson5910 · Answer

oh wait, made a typo

jim_thompson5910 · Answer

sry, forgot something else


doing that will give us 


$$\Large \frac{x^2-16}{(x+4)(\sqrt{x^2+9}+5)}$$


$$\Large \frac{(x+4)(x-4)}{(x+4)(\sqrt{x^2+9}+5)}$$


$$\Large \frac{x-4}{\sqrt{x^2+9}+5}$$


from here, we can use the direct substitution property

jim_thompson5910 · Answer

should be -4/5

jim_thompson5910 · Answer

sry, made a typo earlier, but the work shown above is correct now

anonymous · Answer

oh yeah srry i made a mistake and taht is alright =)