Questions: Evaluate the limit as x approaches 0 from the left of e^(2 / x). If we let t = 2/x, we know that t approaches negative infinity as x approaches 0 from the left. Therefore, by the limit as x approaches negative infinity of e^x = 0, the limit as x approaches 0 from the left of e^(2 / x) = the limit as t approaches negative infinity of e^t = 0

Solution Steps

Let \( t = \frac{2}{x} \). As \( x \rightarrow 0^{-} \), \( t \rightarrow -\infty \).

The original limit \( \lim _{x \rightarrow 0^{-}} e^{2 / x} \) can be rewritten in terms of \( t \) as \( \lim _{t \rightarrow -\infty} e^{t} \).

Using the known limit \( \lim _{t \rightarrow -\infty} e^{t} = 0 \), we conclude that \( \lim _{x \rightarrow 0^{-}} e^{2 / x} = 0 \).

Final Answer