How do you find the limit of #(x ^ 3)(e ^ (-x ^ 2))# as x approaches infinity?

Answer 1

#lim_(x->oo) x^3e^(-x^2) = 0#

Write the limit as:

#lim_(x->oo) x^3e^(-x^2) = lim_(x->oo) x^3/e^(x^2)#
It is now in the indefinite form #oo/oo# and we can apply l'Hospital's rule:
#lim_(x->oo) x^3/e^(x^2) = lim_(x->oo) (d/dx x^3)/(d/dx e^(x^2)) = lim_(x->oo) (3x^2)/(2xe^(x^2)) = lim_(x->oo) (3x)/(2e^(x^2))#

and again:

# lim_(x->oo) (3x)/(2e^(x^2)) = lim_(x->oo) 3/(4xe^(x^2)) = 0#
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Answer 2

To find the limit of (x^3)(e^(-x^2)) as x approaches infinity, we can use the concept of limits.

First, we can rewrite the expression as (x^3)/(e^(x^2)).

As x approaches infinity, the exponential function e^(x^2) grows much faster than any polynomial function, such as x^3.

Therefore, the exponential term dominates the expression, and the limit of (x^3)(e^(-x^2)) as x approaches infinity is 0.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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