Question

sketch the graph of a function that satisfies the conditions given below. f(0)=0, lim f(x)=0 as x→±∞, lim f(x)=lim f(x)=∞ as x→2⁻ and x→ - 2⁺, lim f(x)= - ∞ as x→2⁺ and x→ - 2⁻. choose the correct graph below.

Explanation:

Step1: Analyze \(f(0) = 0\)

The function passes through the origin \((0,0)\).

Step2: Analyze \(\lim_{x

ightarrow\pm\infty}f(x)=0\)
The function has a horizontal - asymptote \(y = 0\) as \(x\) approaches positive and negative infinity.

Step3: Analyze \(\lim_{x

ightarrow2^{-}}f(x)=\lim_{x
ightarrow - 2^{+}}f(x)=\infty\)
The function has vertical asymptotes at \(x = 2\) and \(x=-2\), and the function approaches positive infinity as \(x\) approaches \(2\) from the left and \( - 2\) from the right.

Step4: Analyze \(\lim_{x

ightarrow2^{+}}f(x)=\lim_{x
ightarrow - 2^{-}}f(x)=-\infty\)
The function approaches negative infinity as \(x\) approaches \(2\) from the right and \( - 2\) from the left.

Answer:

A. Option Text (assuming the graph in option A satisfies all the above - mentioned conditions based on the analysis of limits and the point \((0,0)\))