For the function $f(x)$ given below, evaluate $\lim _{x \rightarrow \infty} f(x)$ and $\lim _{x \rightarrow-\infty} f(x)$. \[ f(x)=-2 x+\sqrt{4 x^{2}-3 x} \]
Solution
Step 1 ：The function given is \(f(x) = -2x + \sqrt{4x^2 - 3x}\).
Step 2 ：We are asked to find the limit of this function as \(x\) approaches positive infinity and negative infinity.
Step 3 ：For \(\lim _{x \rightarrow \infty} f(x)\), as \(x\) becomes very large, the \(-2x\) term will become very large and negative, while the \(\sqrt{4x^2 - 3x}\) term will become very large and positive. Since the square root function grows slower than the linear function, the negative term will dominate, and the limit should be negative infinity.
Step 4 ：For \(\lim _{x \rightarrow -\infty} f(x)\), as \(x\) becomes very large negative, the \(-2x\) term will become very large and positive, while the \(\sqrt{4x^2 - 3x}\) term will still be very large and positive. In this case, both terms contribute to the function becoming very large and positive, so the limit should be positive infinity.
Step 5 ：Final Answer: The limit of the function as \(x\) approaches positive infinity is \(\boxed{-\infty}\) and the limit as \(x\) approaches negative infinity is \(\boxed{\infty}\).
