Question

determine whether the following limit is equal to $infty$, $-infty$ or some specific value. $lim_{x
ightarrowinfty}\frac{2x - e^{x}}{4 - 2x}$

Explanation:

Step1: Divide by highest - power of x

Divide both the numerator and denominator by \(x\). The expression \(\lim_{x
ightarrow\infty}\frac{2x - e^{x}}{4 - 2x}\) becomes \(\lim_{x
ightarrow\infty}\frac{2-\frac{e^{x}}{x}}{\frac{4}{x}-2}\).

Step2: Analyze individual limits

We know that \(\lim_{x
ightarrow\infty}\frac{4}{x}=0\) and \(\lim_{x
ightarrow\infty}\frac{e^{x}}{x}=\infty\) (by L'Hopital's rule or the fact that the exponential function \(y = e^{x}\) grows faster than the linear function \(y = x\)).
As \(x
ightarrow\infty\), the numerator \(2-\frac{e^{x}}{x}
ightarrow-\infty\) and the denominator \(\frac{4}{x}-2
ightarrow - 2\).

Step3: Calculate the overall limit

\(\lim_{x
ightarrow\infty}\frac{2-\frac{e^{x}}{x}}{\frac{4}{x}-2}=\frac{-\infty}{-2}=\infty\).

Answer:

\(\infty\)