Question

Find the limit \(L\). Then use the \(\varepsilon-\delta\) definition to prove that the limit is \(L\). $$ \lim _{x \rightarrow 2}(-1) $$

Short answer

The limit of the function \(-1\) as \(x\) approaches 2 is \(-1\). The proof using the \(\varepsilon-\delta\) definition is valid.

Step-by-step solution

Compute the limit

The function is constant and it is -1 for all \(x\). Therefore, the limit as \(x\) approaches to any value is -1. Thus, \(L = -1\).

Use the \(\varepsilon-\delta\) definition

For any \(\varepsilon > 0\), choose \(\delta = 1\) for instance (the choice of \(\delta\) doesn't affect since \(|f(x) - L| = 0\) always holds). Then, if \(0 < |x-2| < \delta\), it follows that \(|-1 - (-1)|= 0 < \varepsilon\), which is always true whatever the \(\varepsilon\) is.

Conclude the results

Based on the results in Step 1 & 2, it is confirmed that \(\lim _{x \rightarrow 2}(-1) = (-1)\) and this complies with the \(\varepsilon-\delta\) definition of a limit. Hence, it is proven that \(L = -1\) is indeed the limit of this function as \(x\) approaches 2.