How do you find #lim (t^3-6t^2+4)/(2t^4+t^3-5)# as #t->oo#?

Answer 1

0

Finding the limit of a function is basically just a way to find out what value we get closer and closer to as we approach a certain number.

Finding the limit at infinity is no different. We should establish a couple of rules before we start.

#...........................barul(| color(white)(----)"Rules"color(white)(----)|).........................#
#color(white)(----)*(oo)/(n) = oocolor(white)(aaaa)"Where n is any integer"#
#color(white)(----)*(n)/(oo) = 0# #.................................................................................................#

Knowing this, we can go ahead and approach the problem as follows:

#barul"|Step 1|"#
The first thing when taking the limit of a rational function is that we should focus our attention on the denominator. There, we must look at the highest power of the polynomial, which in our case is #color(red)[t^4#.
#barul"|Step 2|"#
Next, we will take #color(red)[t^4# and divide it by every term in both the numerator AND the denominator. Doing so we get the following:
#barul"|Step 3|"#

So, in our final step, we look at our rules that we noted above and simplify. Applying our rules, we get the following answer:

#lim_(trarroo)((stackrelcolor(blue)"0"cancel(1/t)-stackrelcolor(blue)"0"cancel(6/t^2)+stackrelcolor(blue)"0"cancel(4/t^4)))/((2+stackrelcolor(blue)"0"cancel(1/t)-stackrelcolor(blue)"0"cancel(5/t^4)))# #color(white)(aaa)#
#lim_(trarroo)(color(blue)0-color(blue)0+color(blue)0)/(2+color(blue)0-color(blue)0)# #color(white)(aaa)#
#lim_(trarroo)(0)/(2)# #color(white)(aaa)#
#color(magenta)[ 0#
#"Answer":color(magenta)(0#
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Answer 2

To find the limit as t approaches infinity of the expression (t^3-6t^2+4)/(2t^4+t^3-5), we need to determine the behavior of the expression as t becomes very large.

First, we observe that the highest power of t in the numerator and denominator is t^4. Dividing both the numerator and denominator by t^4, we get (1/t + -6/t^2 + 4/t^4)/(2 + 1/t - 5/t^4).

As t approaches infinity, the terms with 1/t and 1/t^2 in the numerator become negligible compared to the higher powers of t. Similarly, the terms with 1/t and 1/t^4 in the denominator also become negligible compared to the constant term 2.

Therefore, the expression simplifies to (0 + 0 + 4/t^4)/(2 + 0 - 0), which further simplifies to 4/t^4 divided by 2.

As t approaches infinity, 4/t^4 approaches 0, and dividing 0 by 2 gives us the final result of 0.

Hence, the limit of (t^3-6t^2+4)/(2t^4+t^3-5) as t 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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