Find and simplify limits using L'HopitALS Rule

[EETman]

Lim
X->Infinity

(2e^3x+3x)/(4e^3x+5x)

I know the first step of verifying the ability to use this rule is plugging in a number in order to see if it is of an indeterminate form, however i dont know how to plug in Infinity...

 

[Romsek]

Lim
X->Infinity

(2e^3x+3x)/(4e^3x+5x)

I know the first step of verifying the ability to use this rule is plugging in a number in order to see if it is of an indeterminate form, however i dont know how to plug in Infinity...

there's no magic to it, just plug it in.

for example 2 e^3x -> infinity as x -> infinity

so does 3x

The fact that that info is pretty useless is why L'Hopital's rule is useful. Apply it, you'll see.

 

[EETman]

so the answer then would be as x approaches infinity the limit is positive infinity?

 

[Romsek]

so the answer then would be as x approaches infinity the limit is positive infinity?

No. You've got a quotient. So if you blindly try to plug in infinity you end up with infinity / infinity = undefined.

Thus you need to use L'Hopital's rule to obtain a formula for a limit that's usable.

you've got f(x) = (2e^3x+3x)/(4e^3x+5x), here both numerator and denominator go to infinity as x goes to infinity. That tells you nothing.

so apply L'Hopital's rule by taking the derivative of numerator and denominator and again looking at the quotient

d/dx (2e^3x+3x)/d/dx(4e^3x+5x) = 6e^3x+3/(12e^3x+5), again both the numerator and denominator head towards infinity so we repeat

d/dx (6e^3x+3)/d/dx(12e^3x+5) = 18e^3x/(36e^3x), now there is something to work with. Both numerator and denominator go to infinity but they cancel out as well.

18e^3x/(36e^3x )= 1/2 which is your desired limit.

As a check you can notice that the exponential term grows much faster than the linear term. So as x goes to infinity the exponential terms dominate in both numerator and denominator. Ignoring the linear terms we get f(x) ~ 2e^3x/(4e^3x​) = 1/2

 

[lookagain]

d/dx (2e^3x+3x)/d/dx(4e^3x+5x) = (6e^3x+3)/(12e^3x+5), again both the numerator and denominator head towards infinity so we repeat

d/dx (6e^3x+3)/d/dx(12e^3x+5) = 8e^3x/(36e^3x),now there is something to work with. Both numerator and denominator go to infinity but they cancel out as well.

18e^3x/(36e^3x)= 1/2 which is your desired limit.

As a check you can notice that the exponential term grows much faster than the linear term. So as x goes to infinity the exponential terms dominate in both numerator and denominator. Ignoring the linear terms we get f(x) ~ 2e^3x/(4e^3x)= 1/2

Romsek, don't leave out required grouping symbols, as suggested above in bold.