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

Answer 1

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

We seek:

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

We can manipulate the limits as follows:

# L = lim_(x rarr oo) (3^x-2^x+1)/(4*3^x-2^x-1) * (1/3^x)/(1/3^x) #
# \ \ = lim_(x rarr oo) ((1/3^x)(3^x-2^x+1))/((1/3^x)(4*3^x-2^x-1)) #
# \ \ = lim_(x rarr oo) (3^x/3^x-2^x/3^x+1/3^x)/(4*3^x/3^x-2^x/3^x-1/3^x) #
# \ \ = lim_(x rarr oo) (1-(2/3)^x+(1/3)^x)/(4*1-(2/3)^x-(1/3)^x) #
Now #a^x rarr 0# as #x rarr oo# provided #|a| lt 1#, And so:
# L = (1-0+0)/(4-0-0) #
# \ \ = 1/4 #
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Answer 2

To find the limit as x approaches infinity, we can analyze the behavior of the numerator and denominator separately.

As x approaches infinity, the term 3^x grows much faster than 2^x and 1. Therefore, the term 2^x+1 becomes negligible compared to 3^x.

Similarly, in the denominator, the term 43^x grows much faster than 2^x and 1. Thus, the term 2^x-1 becomes negligible compared to 43^x.

By simplifying the expression, we can rewrite it as (3^x/3^x) / (4*3^x/3^x).

Simplifying further, we get 1 / 4.

Therefore, the limit of (3^x-2^x+1)/(4*3^x-2^x-1) as x approaches infinity is 1/4.

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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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