Answer
(a) Quotient Rule.
(b) Numerator: Difference Rule - Denominator: Power Rule
(c) Numerator: Constant Multiple Rule - Denominator: Sum Rule
Work Step by Step
(a) $$\lim_{x\to0}\frac{2f(x)-g(x)}{(f(x)+7)^{2/3}}=\frac{\lim_{x\to0}(2f(x)-g(x))}{\lim_{x\to0}(f(x)+7)^{2/3}}$$
So from $\lim_{x\to c}\frac{A}{B}$ to $\frac{\lim_{x\to c} A}{\lim_{x\to c} B}$. This is Quotient Rule.
(b) $$\frac{\lim_{x\to0}(2f(x)-g(x))}{\lim_{x\to0}(f(x)+7)^{2/3}}=\frac{\lim_{x\to0}2f(x)-\lim_{x\to0}g(x)}{(\lim_{x\to0}(f(x)+7))^{2/3}}$$
- Numerator: $\lim_{x\to c}(A-B)=\lim_{x\to c}A-\lim_{x\to c}B$. This is Difference Rule.
- Denominator: $\lim_{x\to c}(A^n)=(\lim_{x\to c}A)^n$. This is Power Rule.
(c) $$\frac{\lim_{x\to0}2f(x)-\lim_{x\to0}g(x)}{(\lim_{x\to0}(f(x)+7))^{2/3}}=\frac{2\lim_{x\to0}f(x)-\lim_{x\to0}g(x)}{(\lim_{x\to0}f(x)+\lim_{x\to0}7)^{2/3}}$$
- Numerator: Take a look at $\lim_{x\to0}2f(x)=2\lim_{x\to0}f(x)$. This is Constant Multiple Rule.
- Denominator: $\lim_{x\to c}(A+B)=\lim_{x\to c}A+\lim_{x\to c}B$. This is Sum Rule.
