Questions: Find a function that satisfies the given conditions and sketch its graph. lim x→−∞ h(x)=4, lim x→∞ h(x)=-4, lim x→0− h(x)=4, and lim x→0+ h(x)=-4

Solution Steps

The function \( h(x) \) should approach 4 as \( x \) approaches \(-\infty\) and approach -4 as \( x \) approaches \(\infty\). Additionally, as \( x \) approaches 0 from the left (\(0^-\)), \( h(x) \) should approach 4, and as \( x \) approaches 0 from the right (\(0^+\)), \( h(x) \) should approach -4.

A function that satisfies these conditions is: \[ h(x) = \frac{8x}{x^2 + 1} \] This function approaches 4 as \( x \to -\infty \), -4 as \( x \to \infty \), 4 as \( x \to 0^- \), and -4 as \( x \to 0^+ \).

Final Answer