Question

which of the following describes the end behavior of f(x)=2x/3x^2-3​

Answer 1

Answer:

Step-by-step explanation:

Next time, please enclose the quantity in the denom. inside parentheses, to eliminate any ambiquity regarding what the divisor is.

              2x               2(x)

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

          3x² - 3            3(x - 1)(x + 1)

This is the same as:

2         x

--- * ------------

3     (x-1(x+1)

As x grows large, the 2nd fraction approaches 1/x as a limit, which goes to zero.  Hence, the end behavior involves the x-axis as the horizontal asymptote.  This agrees with the first answer choice.  

Answer 2

The graph approaches 0 as x approaches infinity as well as x approaches negative infinity for the function $f(x)=\frac{2x}{3x^{2} -3}$. This can be obtained by putting ∞ and -∞ in the f(x).

Which of the following describes the end behavior?

Option 1: When x=∞,

Divide each term by x², $f(x)=\frac{\frac{2}{x^{2} } }{3-\frac{3}{x^{2} } }$

Put x=∞, $\lim_{x \to \infty} f(x)$=0/3-0=0  (∵1/∞=0)

The graph approaches 0 as x approaches infinity.

∴The statement is true.

Option 2: When x= -∞,

Divide each term by x², $f(x)=\frac{\frac{2}{x^{2} } }{3-\frac{3}{x^{2} } }$

Put x= -∞, $\lim_{x \to -\infty} f(x)$=0/3-0=0  

The graph approaches 0 as x approaches negative infinity.

∴The statement is true.

Option 3: It is already solved that graph approaches 0 as x approaches infinity.

∴The statement is false.

Option 4: It is already solved that graph approaches 0 as x approaches infinity.

∴The statement is false.

Hence the graph approaches 0 as x approaches infinity as well as x approaches negative infinity for the function $f(x)=\frac{2x}{3x^{2} -3}$. Therefore option 1 and 2 are true.

Learn more about limits here:

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