How do you find the end behavior of # f(x)=3/x^2#?

Question

Elias Ackerman · Accepted Answer

As #x->0, f(x)->oo#. Also, as #x->+oo or -oo, f(x)->0#.  f(x) has thus a vertical asymptote x=0 and an horizontal asymptote  y=0. This is all about the end behaviour of the function.

Benjamin Achtenberg · Answer

undefined To find the end behavior of the function f(x) = 3/x^2, we examine the behavior of the function as x approaches positive and negative infinity.

As x approaches positive infinity (x → ∞), the term x^2 in the denominator becomes very large. When a very large number is squared, it becomes even larger. Therefore, 3 divided by a very large squared number approaches 0. Thus, the end behavior as x approaches positive infinity is f(x) → 0.

Similarly, as x approaches negative infinity (x → -∞), the term x^2 in the denominator becomes very large. However, since x^2 is squared, even a large negative number becomes positive when squared. Therefore, 3 divided by a very large positive squared number still approaches 0. Thus, the end behavior as x approaches negative infinity is also f(x) → 0.

In summary, the end behavior of the function f(x) = 3/x^2 as x approaches positive or negative infinity is f(x) → 0.