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