Evaluate limx→0sin4xsin2x\lim_{x \to 0} \frac{\sin 4x}{\sin 2x}limx→0sin2xsin4x.
Divide numerator and denominator by xxx: limx→0sin4xxsin2xx\lim_{x \to 0} \frac{\frac{\sin 4x}{x}}{\frac{\sin 2x}{x}}limx→0xsin2xxsin4x Adjust coefficients: =limx→04sin4x4xlimx→02sin2x2x= \frac{\lim_{x \to 0} 4 \frac{\sin 4x}{4x}}{\lim_{x \to 0} 2 \frac{\sin 2x}{2x}}=limx→022xsin2xlimx→044xsin4x Since x→0x \to 0x→0, 4x→04x \to 04x→0 and 2x→02x \to 02x→0. =4(1)2(1)=2= \frac{4(1)}{2(1)} = 2=2(1)4(1)=2
Evaluate limx→01−cosxx2\lim_{x \to 0} \frac{1 - \cos x}{x^2}limx→0x21−cosx.
Analyze lines and planes in three-dimensional space.