[-/1 Points] DETAILS MY NOTES SCALCET9 2.6.037. Find the limit. (If the limit is infinite, enter ' \( \infty \) ' or ' \( -\infty \) ', as appropriate. If the limit does not otherwise exist, enter DNE.) \[ \lim _{x \rightarrow \infty} \frac{3-e^{x}}{3+5 e^{x}} \] ASK YOUR TEACHE

To find the limit as \( x \) approaches infinity for the expression \( \frac{3-e^{x}}{3+5 e^{x}} \), we can simplify it by dividing the numerator and denominator by \( e^{x} \): \[ \lim _{x \rightarrow \infty} \frac{\frac{3}{e^{x}} - 1}{\frac{3}{e^{x}} + 5} \] As \( x \rightarrow \infty\), \( \frac{3}{e^{x}} \rightarrow 0\), leading us to: \[ \frac{0 - 1}{0 + 5} = \frac{-1}{5} \] So, the limit is \( -\frac{1}{5} \). In the fascinating world of calculus, exploring limits like this can reveal fascinating behaviors of functions as they stretch toward infinity. This process is akin to a race, where we determine which part of the function is "winning" as we zoom out to the far reaches of the number line! For those curious about real-world applications, limits are integral in fields like physics and engineering. For example, they help model how a system behaves as it approaches extreme conditions, like how a car's speed settles as it nears the maximum speed limit set by physical laws!