We have f(x)=x-cubicroot(x+1) Verify that f(sqrt3/9 -1) =(-2sqrt3)/9 -1 and calculate the limit of f(x) as x approaches positive infinity? Thanks!

Answer 1

See below

#f(x) = x- root{3}{x+1}#
#f(sqrt{3}/9-1) = sqrt{3}/9-1-root{3}{sqrt{3}/9-1+1}# #=sqrt{3}/9-1-root{3}{{3sqrt3}/27}= sqrt{3}/9-1-sqrt{3}/3 = -{2sqrt{3}}/9-1#
Since #x# grows much faster than #root{3}{x+1}# the limit diverges, or more precisely, does not exist!
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Answer 2

To verify that f(sqrt(3)/9 - 1) = (-2sqrt(3))/9 - 1, we substitute sqrt(3)/9 - 1 into the function f(x) and simplify the expression.

f(x) = x - cubicroot(x + 1)

f(sqrt(3)/9 - 1) = (sqrt(3)/9 - 1) - cubicroot((sqrt(3)/9 - 1) + 1)

Simplifying further,

f(sqrt(3)/9 - 1) = (sqrt(3)/9 - 1) - cubicroot(sqrt(3)/9)

Now, let's calculate the limit of f(x) as x approaches positive infinity.

To find the limit of f(x) as x approaches positive infinity, we need to analyze the behavior of the function as x becomes larger and larger.

The function f(x) = x - cubicroot(x + 1) can be simplified to f(x) = x - (x + 1)^(1/3).

As x approaches positive infinity, the term (x + 1)^(1/3) becomes negligible compared to x.

Therefore, the limit of f(x) as x approaches positive infinity is infinity.

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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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