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Which of the following describes the end behavior of the function f(x) = root(x, 3) - 4 as approaches infinity ?

User Kauppfbi
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1 Answer

3 votes

Answer:1) An operator is missing in your statement. Most likely the right expression is:

Explanation:

2x

f(x) = -------------

3x^2 - 3

So, I will work with it and find the result of each one of the statements given to determine their validiy.

2) Statement 1: The graph approaches 0 as x approaches infinity.

Find the limit of the function as x approaches infinity:

2x

Limit when x →∞ of ------------

3x^2 - 3

Start by dividing numerator and denominator by x^2 =>

2x / x^2 2/x

--------------------------- = ---------------

3x^2 / x^2 - 3 / x^2 3 - 3/x^2

2/∞ 0 0

Replace x with ∞ => ------------ = ------- = ---- = 0

3 - 3/∞ 3 - 0 3

Therefore the statement is TRUE.

3) Statement 2: The graph approaches 0 as x approaches negative infinity.

Find the limit of the function as x approaches negative infinity:

2x

Limit when x → - ∞ of ------------

3x^2 - 3

Start by dividing numerator and denominator by x^2 =>

2x / x^2 2/x

--------------------------- = ---------------

3x^2 / x^2 - 3 / x^2 3 - 3/x^2

2/(-∞) 0 0

Replace x with - ∞ => ------------ = ---------- = ---- = 0

3 - 3/(-∞) 3 - 0 3

Therefore, the statement is TRUE.

4) Statement 3: The graph approaches 2/3 as x approaches infinity.

FALSE, as we already found that the graph approaches 0 when x approaches infinity.

5) Statement 4: The graph approaches –1 as x approaches negative infinity.

FALSE, as we already found the graph approaches 0 when x approaches negative infinity.

User Ankii Rawat
by
7.5k points

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