Question

category: quiz
current learning objective: using arrow notation
question 19 practice similar questions
score: 0 of 1 point
describe the local behavior of ( f(x)=-\frac{3}{x^{2}} )
a as ( x
ightarrow 0^{+}, f(x)
ightarrow-infty ), and as ( x
ightarrow 0^{-}, f(x)
ightarrow infty )
b as ( x
ightarrow 0^{+}, f(x)
ightarrow-infty ), and as ( x
ightarrow 0^{-}, f(x)
ightarrow-infty )
c as ( x
ightarrow 0^{+}, f(x)
ightarrow infty ), and as ( x
ightarrow 0^{-}, f(x)
ightarrow-infty )
d as ( x
ightarrow 0^{+}, f(x)
ightarrow infty ), and as ( x
ightarrow 0^{-}, f(x)
ightarrow infty )
submit answer attempts: 0/2
find this question difficult? do you know you can practice another version of this question?
need help?
mamgpt isnt available for this question, but you can still master it if your instructor enabled practice similar questions, youll get unlimited
ai tutoring is no longer available for this assignment.

Explanation:

Step1: Analyze \(x\to0^{+}\)

When \(x\to0^{+}\), \(x^{2}\to0^{+}\) (since \(x>0\) and approaching \(0\), \(x^{2}\) is positive and approaching \(0\)). Then \(\frac{3}{x^{2}}\to+\infty\). So \(f(x)=-\frac{3}{x^{2}}\to-\infty\).

Step2: Analyze \(x\to0^{-}\)

When \(x\to0^{-}\), \(x^{2}\to0^{+}\) (because \(x < 0\), \(x^{2}=(-|x|)^{2}=|x|^{2}\), and as \(x\to0^{-}\), \(|x|\to0\), so \(x^{2}\to0^{+}\)). Then \(\frac{3}{x^{2}}\to+\infty\). So \(f(x)=-\frac{3}{x^{2}}\to-\infty\).

Answer:

B. as \(x\to0^{+},f(x)\to-\infty\), and as \(x\to0^{-},f(x)\to-\infty\)