Q. 2 - Calculus

Question

The definition of infinite limits and limits at infinity: Write each limit statement that follows in terms of the formal definition of limit. Then approximate the largest value of $\delta or N$corresponding to $\in =0.5 or M=100$, as appropriate, and illustrate this choice of $\delta or N$  on a graph of $f$.

     

   $\underset{x\to \mathrm{\infty }}{lim} \frac{2x}{x-1}=2$

Step-by-Step Solution

Answer

Ans: From the definition of limit

    $$\begin{gathered} \left|\frac{2x}{x-1}-2\right|<ϵwhenever x>N \\ \left|\frac{2x-2x+2}{x-1}\right|<ϵwhenever x>N \\ \left|\frac{2}{x-1}\right|<ϵwhenever x>N \end{gathered}$$

Step 1. Given information.

given,

      $\underset{x\to \mathrm{\infty }}{lim} \frac{2x}{x-1}=2$

Step 2. Solution

 $\begin{array}{r}\underset{x\to m}{lim} \frac{2x}{x-1}=2\\ f\left(x\right)=\frac{2x}{x-1}\end{array}$

From the definition of limit,

   $$\begin{gathered} \left|\frac{2x}{x-1}-2\right|<ϵwhenever x>N \\ \left|\frac{2x-2x+2}{x-1}\right|<ϵwhenever x>N \\ \left|\frac{2}{x-1}\right|<ϵwhenever x>N \end{gathered}$$

Step 3. From the graph, one can say that the above inequality holds true when N &#62; 5 .