directions:\n- provide a solution for every problem, place your final answer in a box.\n- keep your solution…
directions:\n- provide a solution for every problem, place your final answer in a box.\n- keep your solution readable.\n- sharing of materials such as eraser, pens, paper is not allowed.\n- do not disturb others, focus on solving on your own.\n- cheating in any form is prohibited.\ntest i. solve the following:\n1. find the value of \\(\\lim_{x\\to0}\\frac{\\tan^{2}x}{3x})\n2. find the value of \\(\\lim_{x\\to - 2}\\frac{3x^{3}-1x^{2}+4x}{9x^{2}+12x + 4})
Answer
Explanation:
Step1: Analyze the first limit
For $\lim_{x\rightarrow0}\frac{\tan^{2}x}{3x}$, use the fact that $\tan x\sim x$ as $x\rightarrow0$. So $\tan^{2}x\sim x^{2}$ as $x\rightarrow0$. Then $\lim_{x\rightarrow0}\frac{\tan^{2}x}{3x}=\lim_{x\rightarrow0}\frac{x^{2}}{3x}$.
Step2: Simplify the first - limit expression
$\lim_{x\rightarrow0}\frac{x^{2}}{3x}=\lim_{x\rightarrow0}\frac{x}{3}=0$.
Step3: Analyze the second limit
For $\lim_{x\rightarrow0}\frac{3x - 12x^{2}}{9x^{3}-x^{2}+4x}$, factor out the greatest - common factor from the numerator and the denominator. The numerator $3x - 12x^{2}=x(3 - 12x)$ and the denominator $9x^{3}-x^{2}+4x=x(9x^{2}-x + 4)$.
Step4: Simplify the second - limit expression
Cancel out the common factor $x$ (since $x\neq0$ when taking the limit as $x\rightarrow0$), we get $\lim_{x\rightarrow0}\frac{3 - 12x}{9x^{2}-x + 4}$. Then substitute $x = 0$ into the expression, $\frac{3-12\times0}{9\times0^{2}-0 + 4}=\frac{3}{4}$.
Answer:
- $0$
- $\frac{3}{4}$