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How do you find the limit #lim (3^(x+1)-2^(x+4))/(3^(x-2)+2^(x-1)+6)# as #x->oo#?

Answer 1

Isaiah Allen

#3#

#lim_(x->oo) (3^(x+1)-2^(x+4))/(3^(x-2)+2^(x-1)+6) = lim_(x->oo)(3^-1-2^4*(2/3)^x)/(3^-2+2^-1(2/3)^x+6/3^x) = 3#

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Answer 2

Scarlett Allison

To find the limit as x approaches infinity, we can analyze the highest power terms in the numerator and denominator. In this case, the highest power terms are 3^(x+1) and 3^(x-2) in the numerator and denominator respectively.

As x approaches infinity, 3^(x+1) grows much faster than 3^(x-2). Therefore, we can ignore the lower power terms in the numerator and denominator.

By simplifying the expression, we get (3^(x+1))/(3^(x-2)).

Using the properties of exponents, we can rewrite this as 3^((x+1)-(x-2)), which simplifies to 3^3.

Thus, the limit of the given expression as x approaches infinity is 3^3, which is equal to 27.

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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.

How do you find the limit #lim (3^(x+1)-2^(x+4))/(3^(x-2)+2^(x-1)+6)# as #x-&gt;oo#?

How do you find the limit #lim (3^(x+1)-2^(x+4))/(3^(x-2)+2^(x-1)+6)# as #x->oo#?