Which of the following functions grows at the same rate as e^x as x goes to infinity?
a) e^(x+3)
b) e^(3x)
c) e^(2x)
d) e^(-x)
e^-x decreases, so it is clearly not a choice.
All the rest have slopes that grow to infinity, and it makes little sense to compare infinities.
Still,
e^(x+3) = e^3 * e^x
e^(2x) = (e^x)^2
e^(3x) = (e^x)^3
of all those, I'd say that e^(x+3) grows most like e^x, but its slope is always e^3 times as much. Poorly worded question. k*∞ = ∞^2 = ∞^3 = ∞.
Well, well, well, looks like we have some exponential equations to solve! Let me put on my mathematician clown shoes and help you out.
As x goes to infinity, we are looking for the function that also grows at the same rate as e^x. So, if we can rewrite any of these options to match the form of e^x, we'll have our winner!
Let's take a look at each option:
a) e^(x+3): This can be rewritten as e^x * e^3, but hold your horses, that extra e^3 will grow at a faster rate than e^x. So, it's not the one.
b) e^(3x): This one is already in the form e^x! Hurray! As x goes to infinity, this function will indeed grow at the same rate as e^x.
c) e^(2x): Ohh, a sneaky triple agent here! This function will grow at a slower rate than e^x because that 2x inside will make it grow twice as slow. So not this one either.
d) e^(-x): Ah, an inverted exponent! This function will actually shrink as x goes to infinity, since the negative sign makes it flip. So definitely not this one.
So, the correct answer is b) e^(3x). It will grow at the same rate as e^x. Ta-da!
To determine which of the functions grows at the same rate as e^x as x goes to infinity, we can compare the growth rates by looking at the exponent of the base e.
a) e^(x+3) = e^x * e^3
b) e^(3x) = e^x * e^x * e^x = (e^x)^3
c) e^(2x) = e^x * e^x = (e^x)^2
d) e^(-x) = 1/e^x
As x goes to infinity, the functions with higher powers of e^x in their forms will grow at a faster rate.
Therefore, the function that grows at the same rate as e^x as x goes to infinity is option c) e^(2x).
To determine which of the given functions grows at the same rate as e^x as x goes to infinity, we need to compare the growth rates of the functions.
The function e^x grows exponentially as x increases. As x goes to infinity, e^x increases without bound.
Let's examine each option one by one:
a) e^(x+3): The exponent x+3 does not affect the growth rate of e^x. As x goes to infinity, this function still grows exponentially and at the same rate as e^x.
b) e^(3x): In this case, the exponent 3x increases at a faster rate compared to just x. As x goes to infinity, e^(3x) grows even faster than e^x. Therefore, e^(3x) does not grow at the same rate as e^x as x goes to infinity.
c) e^(2x): Similar to the previous case, the exponent 2x increases at a faster rate compared to x. As x goes to infinity, e^(2x) grows faster than e^x. Hence, e^(2x) does not grow at the same rate as e^x as x goes to infinity.
d) e^(-x): In this case, the exponent -x decreases as x goes to infinity. As x approaches infinity, e^(-x) tends to approach zero. Therefore, e^(-x) does not grow at the same rate as e^x as x goes to infinity.
Therefore, the answer is option a) e^(x+3), which grows at the same rate as e^x as x goes to infinity.