Chapter 4.7, Problem 3E

(a)

To determine

To prove:If $\underset{x\to a}{\lim }f\left(x\right)=L$ and $\underset{x\to a}{\lim }g\left(x\right)=M$ then, $\underset{x\to a}{\lim }\left(f+g\right)\left(x\right)=L+M$ .

(b)

To determine

To prove: If $\underset{x\to a}{\lim }f\left(x\right)=L$ and $\underset{x\to a}{\lim }g\left(x\right)=M$ then, $\underset{x\to a}{\lim }\left(f-g\right)\left(x\right)=L-M$

(c)

To determine

To prove: If $\underset{x\to a}{\lim }f\left(x\right)=L$ and $\underset{x\to a}{\lim }g\left(x\right)=M$ then, $\underset{x\to a}{\lim }\left(fg\right)\left(x\right)=LM$

(d)

To determine

To prove: If $\underset{x\to a}{\lim }f\left(x\right)=L$ and $\underset{x\to a}{\lim }g\left(x\right)=M$ then, $\underset{x\to a}{\lim }rf\left(x\right)=rL$ .

(e)

To determine

To prove: If $\underset{x\to a}{\lim }f\left(x\right)=L$ and $\underset{x\to a}{\lim }g\left(x\right)=M$ then, $\underset{x\to a}{\lim }\left(\frac{f}{g}\right)\left(x\right)=\frac{L}{M}$

(f)

To determine

To prove: $\underset{x\to a}{\lim }{x}^n=a^n$ for all $n\in N$ .