Q. 48 - Calculus

Question

Write delta-epsilon proofs for each of the limit statements $\underset{x\to c}{\lim }f\left(x\right)=L$ in Exercises $47-60$.

$\underset{x\to 2}{\lim }\left(3-4x\right)=-5$.

Step-by-Step Solution

Answer

Delta-epsilon proof for $\underset{x\to 2}{\lim }\left(3-4x\right)=-5$ is, whenever $0<\left|x-2\right|<\delta$, we also have $\left|\left(3-4x\right)+5\right|<\epsilon$.

Step 1 . Given information

$\underset{x\to 2}{\lim }\left(3-4x\right)=-5$.

Step 2 . Consider &#949; &#62; 0 , choose &#948; = &#949; 4 .

For all $x$ with $0<\left|x-2\right|<\delta$,we also have $\left|\left(3-4x\right)+5\right|<\epsilon$.

$$\begin{gathered} \left|\left(3-4x\right)+5\right|=\left|3-4x+5\right| \\ =\left|8-4x\right| \\ =4\left|2-x\right| \\ =4\left|x-2\right| \\ <4\delta \\ =4\left(\frac{\epsilon }{4}\right) \\ =\epsilon \end{gathered}$$

Therefore, whenever $0<\left|x-2\right|<\delta$, we also have $\left|\left(3-4x\right)+5\right|<\epsilon$.