Comparing Degrees in Polynomial Limits
Determine the limit as \(x \to \infty\) of the function \(f(x) = \frac{2x^3 – 5x + 7}{3x^3 + 4x^2 – 2}\). Explain your reasoning.
Part d
Describe the behavior of the locus of \( z(t) \) as \( t \) approaches \( +\infty \) and as \( t \) approaches \( -\infty \). Explain what happens to the spiral in these limiting cases.
Asymptotic Derivative Analysis for Exponential and Rational Functions
Consider the function \(f(x)=e^x+\frac{5}{x}\). (a) Differentiate \(f(x)\) with respect to \(x\). (b) Determine the dominant term in \(f'(x)\) as \(x \to \infty\) and explain its significance in terms of the function’s behavior.