Evaluate $\lim_{x \to 0} \frac{\sin(5x) - 5x}{x^3}$.
Using Taylor series for $\sin(5x) = 5x - (5x)^3/6 +$... Thus ($\sin(5x) - 5x)/x^3 = -125/6 = -20.8333$... Rounding to -20.83 or -20.833.
What is the value of $\lim_{x \to 0} \frac{e^x - 1 - x}{x^2}$?
Using L'Hopital's rule twice: $\lim (e^x - 1)/(2x) = \lim e^x / 2 = 1/2 = 0.5$.
Find $\lim_{x \to \infty} \left(1 + \frac{3}{x}\right)^{2x}$.
Standard limit form: $\lim (1 + a/x)^{bx} = e^{ab}$. Here a=3, b=2, so limit = $e^6$.
Find the value of $\lim_{x \to 0} \frac{\ln(1 + 4x)}{\tan(2x)}$.
Using L'Hopital's rule or standard limits: $\lim_{x \to 0} \frac{\ln(1+4x)}{4x} \cdot \frac{2x}{\tan(2x)} \cdot \frac{4}{2} = 1 \cdot 1 \cdot 2 = 2$.
Consider f(x) = |x - 2| + |x - 5|. At how many points in $\mathbb{R}$ is f(x) NOT differentiable?
f(x) has sharp corners at x = 2 and x = 5, so it is non-differentiable at exactly 2 points.
