Proving a limit as X approaches infinity

[ssmmss]

Hello,

Please let me know if my logic and arithmetic are correct. I am trying to prove that lim x^4/(e^3x) =0, as X approaches infinity

Solution:

The limit of x^4 over e^3x is the same as the limit of the derivative of x^4 over e^3x.

lim x^4/e^3x = lim (〖(x〗^4)')/(〖(e〗^3x)')= lim 〖4x〗^3/〖3e〗^3x = lim (〖(4x〗^3)')/(〖(3e〗^3x)')= lim 〖12x〗^2/〖9e〗^3x = lim ((〖12x〗^2)')/(〖(9e〗^3x)')= lim (24x^ )/〖27e〗^3x = lim ((24〖x)'〗^ )/((〖27e〗^3x)')= lim 〖24〗^ /〖81e〗^3x =0

Thanks in advance.

 

[pka]

Please let me know if my logic and arithmetic are correct. I am trying to prove that lim x^4/(e^3x) =0, as X approaches infinity Solution:
The limit of x^4 over e^3x is the same as the limit of the derivative of x^4 over e^3x.

lim x^4/e^3x = lim (〖(x〗^4)')/(〖(e〗^3x)')= lim 〖4x〗^3/〖3e〗^3x = lim (〖(4x〗^3)')/(〖(3e〗^3x)')= lim 〖12x〗^2/〖9e〗^3x = lim ((〖12x〗^2)')/(〖(9e〗^3x)')= lim (24x^ )/〖27e〗^3x = lim ((24〖x)'〗^ )/((〖27e〗^3x)')= lim 〖24〗^ /〖81e〗^3x =0

Your process is correct. BUT you need to work on notation.

 

[ssmmss]

Your process is correct. BUT you need to work on notation.

Thank you, what did you mean by notation?

 

[pka]

Thank you, what did you mean by notation?

If you mean to write $\displaystyle 3e^{3x}$, then it is incorrect to write $\displaystyle [3e]^{3x}$.

 

[ssmmss]

If you mean to write $\displaystyle 3e^{3x}$, then it is incorrect to write $\displaystyle [3e]^{3x}$.

Thanks. The copy and past didn't work neatly. How about now?

lim x^4/e^3x = lim (x^4)'/(e^3x)'= lim 4x^3/3e^3x = lim (4x^3)'/(3e^3x)'= lim 12x^2/9e^3x = lim (12x^2)'/(9e^3x)'= lim 24x/(27e^3x) = lim (24x)'/(27e^3x)'= lim 24 /81e^3x =0

 

[mmm4444bot]

The Order of Operations tells us that exponentiation is done before multiplication.

Therefore, the expression e^3x means e^3 times x.

If you want to change the Order of Operations (i.e., do the multiplication first), then you need to type grouping symbols around the exponent.

e^(3x) means multiply 3 times x first, and then raise e to the power of 3x.

:cool:

 

[ssmmss]

The Order of Operations tells us that exponentiation is done before multiplication.

Therefore, the expression e^3x means e^3 times x.

If you want to change the Order of Operations (i.e., do the multiplication first), then you need to type grouping symbols around the exponent.

e^(3x) means multiply 3 times x first, and then raise e to the power of 3x.

:cool:

Thank you both, you've been very helpful.

 

[HallsofIvy]

Also, you say

The limit of x^4 over e^3x is the same as the limit of the derivative of x^4 over e^3x.

which is not true. What you meant to say was "the limit of x^3 over e^3x is the same as the limit of the derivative of x^3 over the limit of the derivative of e^3x.

 

[lookagain]

[ . . . ] What you meant to say was "the limit of x^3 over e^3x is the same as the limit of the derivative of x^3 over
the limit of the derivative of e^3x."

In this thread, we have also been emphasizing to the student that grouping symbols are needed in certain places,
such as around the exponent for the base e, as in "e^(3x)."



I would further amend it to (leaving in the bold letters from HallsofIvy):

"The limit of the quotient of x^3 and e^(3x) is the same as the limit of the quotient of the derivative of x^3
and the limit of the derivative of e^(3x)."