20. [-/1 Points] DETAILS MY NOTES SCALCET9 2.6.AE.007. EXAMPLE 7 Video Example 4\( ) \) Evaluate \( \lim _{x \rightarrow 0^{-}} e^{8 / x} \). SOLUTION If we let \( t=\frac{8}{x} \), we know that \( t \rightarrow-\infty \) as \( x \rightarrow 0^{-} \). Therefore, by \( \lim _{x \rightarrow-\infty} e^{x}=0 \), \( \lim _{x \rightarrow 0^{-}} e^{8 / x}=\lim _{t \rightarrow-\infty} e^{t}=\square \) (See this exercise.)

As \(x\) approaches \(0^{-}\), \(e^{8/x}\) can seem pretty daunting at first! When \(x\) is negative and close to zero, \(8/x\) becomes a large negative number, driving the limit toward \(e^{-\infty}\). And we know from our studies that any exponential function with a base greater than 1 will approach 0 as its exponent moves toward negative infinity. So, the final answer is that this limit evaluates to \(0\). If you'd like to explore further, consider looking at the behavior of other exponential limits, such as \(\lim_{x \to 0^+} e^{8/x}\), to see the contrast. This way, you'll gain a deeper appreciation of how small changes in input can drastically change the output of exponential functions! Plus, it's a fantastic way to reinforce your understanding of limits and their interpretations in various quadrants.