Q. 89 - Calculus

Question

Prove that the sum rule for limits also applies for limits as $x\to \infty$ : If $\underset{x\to \infty }{\lim }f\left(x\right)=L$ and $\underset{x\to \infty }{\lim }g\left(x\right)=M$, then $\underset{x\to \infty }{\lim }\left(f\right(x)+g(x\left)\right)=L+M.$

Step-by-Step Solution

Answer

It is proved that  If $\underset{x\to \infty }{\lim }f\left(x\right)=L$ and $\underset{x\to \infty }{\lim }g\left(x\right)=M$, then $\underset{x\to \infty }{\lim }\left(f\right(x)+g(x\left)\right)=L+M.$

Step 1. Given Information

We are given two functions $f\left(x\right) \text{and} g\left(x\right)$. 

Step 2. Proving the statement

Given $\epsilon >0$, we can choose $N_1>0$ to get $f\left(x\right)$ within $\frac{\epsilon }{2}$ of L and also choose $N_2>0$ to get $g\left(x\right)$ within $\frac{\epsilon }{2}$ of M,

Then for $N=\max \left(N_1,N_2\right)$ and $x>N$ we have,

$(L+M)-\epsilon <f\left(x\right)+g\left(x\right)<(L+M)+\epsilon$

Hence, the sum rule for limits also applies for limits as $x\to \infty$. If $\underset{x\to \infty }{\lim }f\left(x\right)=L$ and $\underset{x\to \infty }{\lim }g\left(x\right)=M$, then $\underset{x\to \infty }{\lim }\left(f\right(x)+g(x\left)\right)=L+M$.