Question

Which of the following describes the end behavior of f(x) = 2x 3x2 − 3 ? The graph approaches 0 as x approaches infinity. The graph approaches 0 as x approaches negative infinity. The graph approaches 2/3 as x approaches infinity. The graph approaches –1 as x approaches negative infinity.

Answer 1

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

               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.

Answer 2

The end behavior of a function is how the function approaches infinity or negative infinity.

The end behavior of the function is (a) the graph approaches 0 as x approaches infinity

The function is given as:

$\mathbf{f(x) = \frac{2x}{3x^2 - 3}}$

Take the limit of the function to infinity

$\mathbf{f(x) \lim_{x \to \infty} = \frac{2x}{3x^2 - 3}}$

So, we have:

$\mathbf{f(x) \lim_{x \to \infty} = \frac{2x/x^2}{3x^2/x^2 - 3/x^2}}$

Simplify

$\mathbf{f(x) \lim_{x \to \infty} = \frac{2/x}{3 - 3/x^2}}$

Substitute infinity for x

$\mathbf{f(x) \lim_{x \to \infty} = \frac{2/(\infty)}{3 - 3/(\infty)^2}}$

Simplify

$\mathbf{f(x) \lim_{x \to \infty} = \frac{0}{3 - 3/\infty}}$

$\mathbf{f(x) \lim_{x \to \infty} = \frac{0}{3 - 0}}$

$\mathbf{f(x) \lim_{x \to \infty} = \frac{0}{3}}$

$\mathbf{f(x) \lim_{x \to \infty} = 0}$

This means that;

As x approaches infinity, the function approaches 0.

Hence, the correct option is (a)

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