Answer to Question #342830 in Calculus for Ddeonu

Question

Problem 1: Use the tabular method to determine if the limits of the following functions exist:

a) lim𝑥→3 2/(𝑥−3)^2

b) lim𝑥→3 2/(𝑥−3)^3

Answer 1

a)

$$\lim\limits_{x\to 3}\dfrac{2}{(x-3)^2}$$ $$\def\arraystretch{1.5} \begin{array}{c:c} x & f(x) \\ \hline 2.9 & 200 \\ \hdashline 2.99 & 20000 \\ \hdashline 2.999 & 2000000 \\ \hdashline 2.9999 & 200000000 \\ \hdashline 3.0001 & 200000000 \\ \hdashline 3.001 & 2000000 \\ \hdashline 3.01 & 20000 \\ \hdashline 3.1 & 200 \\ \hdashline \end{array}$$

$$\lim\limits_{x\to 3}\dfrac{2}{(x-3)^2}=\infin$$

b)

$$\lim\limits_{x\to 3}\dfrac{2}{(x-3)^3}$$ $$\def\arraystretch{1.5} \begin{array}{c:c} x & f(x) \\ \hline 2.9 & -2000 \\ \hdashline 2.99 & -2000000 \\ \hdashline 2.999 & -2000000000 \\ \hdashline 2.9999 & -2000000000000 \\ \hdashline 3.0001 & 2000000000000 \\ \hdashline 3.001 & 2000000000 \\ \hdashline 3.01 & 2000000 \\ \hdashline 3.1 & 2000 \\ \hdashline \end{array}$$

$$\lim\limits_{x\to 3}\dfrac{2}{(x-3)^3}=DNE(\text{does not exist})$$