# Chapter 2 - Section 2.2 - Limit of a Function and Limit Laws - Exercises - Page 67: 51

(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.

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