Question

How would I solve the equation limit of 4x^2/x-2 while x tends to ∞? I can solve limits with numbers, but I wasn't sure what to do in this situation. Would I just eliminate?

Answer 1

is that 4x^2/(x-2) or 4x^2/x-2 ?

Answer 2

\[ \text{ I believe you meant } \lim_{x \rightarrow \infty} 4x^2/(x-2) \\ \text{ so let's look at } \lim_{x \rightarrow \infty} \frac{4x^2}{x-2}\]

Answer 3

Let me ask you if you know the following answers:
\[\lim_{x \rightarrow \infty} 4x^2= ? \\ \lim_{x \rightarrow \infty}(x-2)=?\]

Answer 4

lim if 4x^2 = ∞
lim of (x-2) = ∞

Answer 5

right
do you think one function gets bigger faster

Answer 6

I'm talking about comparing f(x)=4x^2 to g(x)=x-2
and telling me if you think one takes off way faster in getting bigger

Answer 7

I'm just guessing here, but would it be the function f(x) = 4x^2?

Answer 8

because the function is squared?

Answer 9

yep you can also look at the graph there
the 4x^2 is way bigger than the x-2 for really big values of x

Answer 10

that means you will be dividing really big numbers by other numbers that are less big

Answer 11

what do you think that means about the limit to your initial question?

Answer 12

the limit will go to infinity

Answer 13

right
if the 4x^2 was on bottom and the x-2 was on top than you would say 0 instead

Answer 14

Ok. I get it now

Answer 15

Thank You!

Answer 16

if you had the same degree though then you would need to do something differently

Answer 17

same degree means the top and bottom has the same largest exponent

Answer 18

for example
\[\lim_{n \rightarrow \infty} \frac{4x^2}{x^2-2}=\frac{4}{1}=4 \\ \text{ I just stole the coefficients of } x^2 \text{ on top and on bottom }\]
coefficient of 4x^2 is 4
coefficient of x^2 is 1

Answer 19

oops that was suppose to say x approaches infinity
not n

Answer 20

same degree examples:
\[\lim_{x \rightarrow \infty} \frac{4x^2}{x^2-1}=\frac{4}{1}=4 \\ \lim_{x \rightarrow
\infty} \frac{4x^3}{x^2-x^3} =\frac{4}{-1}=-4 \\ \lim_{x \rightarrow \infty} \frac{5x-1}{6x-2}=\frac{5}{6} \]

Answer 21

bigger degree on top examples:
\[\lim_{x \rightarrow \infty} \frac{4x^3}{x-1}=\infty \\ \lim_{x \rightarrow \infty} \frac{x^4-1}{4x^2+1}=\infty\]

Answer 22

bigger degree on bottom examples:
\[\lim_{x \rightarrow \infty} \frac{4x}{2x^3-1}=0 \\ \lim_{x \rightarrow \infty} \frac{4x^3+5x^2+1}{8x^4}=0 \\ \lim_{x \rightarrow \infty} \frac{1}{x}=0\]

Answer 23

notice all of these examples I mentioned are when you have a polynomial over a polynomial and x is approaching infinity

Answer 24

oops one more thing about the bigger degree on top examples

Answer 25

I chose ones that gave you positive infinity

Answer 26

let me choose someones that would give you negative infinity instead

Answer 27

I got the answer for my question, but how does an answer become 0 or a number besides infinity?

Answer 28

I'm not sure I understand how you got the answers for the examples

Answer 29

add to this above to my section on
the degree on top is bigger examples:
\[\lim_{x \rightarrow \infty} \frac{4x^3}{x-1}=\infty \\ \lim_{x \rightarrow \infty} \frac{x^4-1}{4x^2+1}=\infty \\ \lim_{x \rightarrow \infty } \frac{-x^4-1}{4x^2+1}=-\infty \\ \lim_{x \rightarrow \infty} \frac{x^4-1}{-4x^2+1}= - \infty \\ \lim_{x \rightarrow \infty} \frac{-x^4-1}{-4x^2+1}=\infty\]

Answer 30

did you understand my very first examples before beginning the other same degree examples

Answer 31

this one:
\[\lim_{n \rightarrow \infty} \frac{4x^2}{x^2-2}=\frac{4}{1}=4 \\ \text{ I just stole the coefficients of } x^2 \text{ on \top and on bottom } \]

coefficient of 4x^2 is 4
coefficient of x^2 is 1

Answer 32

yes

Answer 33

I understood that one

Answer 34

\[\lim_{x \rightarrow \infty} \frac{4x^2}{x^2-1}=\frac{4}{1}=4 \\ \lim_{x \rightarrow \infty} \frac{4x^3}{x^2-x^3} =\frac{4}{-1}=-4 \\ \lim_{x \rightarrow \infty} \frac{5x-1}{6x-2}=\frac{5}{6} \]
which of these same degree examples did you understand if any

Answer 35

the first one. Do you just take the coefficients?

Answer 36

the coefficients of the terms with the largest exponents on top and bottom
this is only if the largest exponents match that you can do this

Answer 37

the first example deg(4x^2)=deg(x^2-1) so you can apply this and say the limit is 4/1 or 4
the second example deg(4x^3)=deg(-x^3+x^2) so you can apply this and say the limit is 4/(-1) or -4
the third example degree(5x-1)=deg(6x-2) so you can apply this and say the limit is 5/6

Answer 38

further information if you need it:
deg(4x^2)=deg(x^2-1)=2
deg(4x^3)=deg(-x^3+x^2)=3
deg(5x-1)=deg(6x-2)=1

Answer 39

so these are the limits when you have a polynomial over a polynomial as x approaches infinity or negative infinity:

if the deg(top)=deg(bottom) then the answer is going to come from the coefficients of the terms with largest exponents

if the deg(top)>deg(bottom) then the answer is going to be either -infinity or +infinity

if the deg(top)<deg(bottom) then the answer is going to be zero

Answer 40

deg( ) just means the degree of the polynomial inside the deg( )

Answer 41

ok

Answer 42

I think i get it

Answer 43

do you know this limit then?
\[\lim_{x \rightarrow \infty} \frac{-9x^2+5x-1}{3x^2}\]

Answer 44

3

Answer 45

almost

Answer 46

the coefficient of -9x^2 is -9

Answer 47

so your answer should be -9/3 and if reduced just -3

Answer 48

ok

Answer 49

how do I determine the degree of the polynomial again?

Answer 50

it is largest exponent

for examples:
deg(5x^3-4x+1)=3
deg(2x-1)=1
deg(5)=0
deg(6x^2-1)=2

Answer 51

now I remember

Answer 52

sorry

Answer 53

np
all of this was probably learned years before you entered this
calculus class
I don't even think it takes very much time to forget

Answer 54

I think I got it. Thank You!

Answer 55

http://tutorial.math.lamar.edu/Classes/CalcI/LimitsAtInfinityI.aspx
this contains more examples
and it goes into more detail using the following limit to evaluate other limits
\[\lim_{x \rightarrow \infty} \frac{1}{x^r}=0 \text{ where } r \text{ is a positive rational number }\]

Answer 56

the above things i mentioned were shortcuts

Answer 57

the site there mentions the non short cut way is what i'm trying to say

Answer 58

i will be back later if you have more questions