lim (x*3^x + (lnx)^4)/((1+4x)*3^x + x^17)
x-->infinity
explain how you do
x-->infinity
explain how you do
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It helps to know some properties of limits, especially that limits of combinations of functions are usually the same as the combinations of the limits of the functions, but you have to know the exceptions when you can't just do that. Those are called indeterminate forms.
It should be obvious that x(3^x) is much bigger than (ln x)^4.
Also it looks like x^17 might be kind of big.
Really it's nothing compared to 3^x. Do you need me to show you that?
If you were asking a simple question like what is the limit of x*(3^x)/((lnx)^4) then I would do that.
The algebraic form of that process is to multiply the top and bottom of this monstrous expression by (1/3)^x and that will separate the children from the adults
lim (x + [(lnx)^4/(3^x)]) / ((1+4x) + [(x^17)/[x^3])
x--> infinity
By applying L'Hopital's Rule, sandwich theorem, and probably some other stuff to avoid applying L'Hopital's Rule 17 times, you can see it equals this
lim (x / (1 + 4x))
x--> infinity
The answer is 1/4
It's more evident again if it's simplified to
lim (1 / (4 + 1/x))
x--> infinity
It should be obvious that x(3^x) is much bigger than (ln x)^4.
Also it looks like x^17 might be kind of big.
Really it's nothing compared to 3^x. Do you need me to show you that?
If you were asking a simple question like what is the limit of x*(3^x)/((lnx)^4) then I would do that.
The algebraic form of that process is to multiply the top and bottom of this monstrous expression by (1/3)^x and that will separate the children from the adults
lim (x + [(lnx)^4/(3^x)]) / ((1+4x) + [(x^17)/[x^3])
x--> infinity
By applying L'Hopital's Rule, sandwich theorem, and probably some other stuff to avoid applying L'Hopital's Rule 17 times, you can see it equals this
lim (x / (1 + 4x))
x--> infinity
The answer is 1/4
It's more evident again if it's simplified to
lim (1 / (4 + 1/x))
x--> infinity