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

Transcript text: Find a function that satisfies the given conditions and sketch its graph. \[ \lim _{x \rightarrow-\infty} h(x)=4, \lim _{x \rightarrow \infty} h(x)=-4, \lim _{x \rightarrow 0^{-}} h(x)=4, \text { and } \lim _{x \rightarrow 0^{+}} h(x)=-4 \]

Solution

Solution Steps

Step 1: Identify the behavior of the function at different limits

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.

Step 2: Construct a function that satisfies the given limits

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

The function $ h(x) = \frac{8x}{x^2 + 1} $ satisfies the given conditions.

{"axisType": 3, "coordSystem": {"xmin": -10, "xmax": 10, "ymin": -5, "ymax": 5}, "commands": ["y = (8x)/(x^2 + 1)"], "latex_expressions": ["$y = \\frac{8x}{x^2 + 1}$"]}

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