Graph of the Function $y = \dfrac{e^x}{x}$

$y = \dfrac{e^x}{x}$ is an exponential divided by a linear function. Unlike $\dfrac{\ln x}{x}$, here the numerator dominates overwhelmingly, so the values climb ever more steeply to the right. The graph splits in two at the $y$-axis.

Domain and sign

The domain is $x \neq 0$. Since $e^x$ is always positive, the sign is decided by the denominator alone: $y > 0$ for $x > 0$ and $y < 0$ for $x < 0$, so the curve occupies the first and third quadrants.

Monotonicity and extrema

The quotient rule gives the derivative.

Both $e^x$ and $x^2$ are positive, so the sign is that of $x - 1$. The function decreases on $x < 0$ and on $0 < x < 1$, then increases for $x > 1$, giving a local minimum of $e$ at $x = 1$. The negative branch has no extremum and simply decreases throughout.

Range

Together the range is $y < 0$ or $y \geq e$; no value in $0 \leq y < e$ is ever attained.

Asymptotes

The $y$-axis is a vertical asymptote, approached toward $+\infty$ from the right and toward $-\infty$ from the left. As $x \to -\infty$ the factor $e^x$ collapses to $0$ and $y \to 0^{-}$, so the $x$-axis is a horizontal asymptote on the left only. There is none on the right, because $e^x$ outgrows every polynomial.

Concavity

The second derivative is $y'' = \dfrac{e^x(x^2 - 2x + 2)}{x^3}$. Since $x^2 - 2x + 2 = (x - 1)^2 + 1$ is always positive, the sign is that of $x^3$: concave up for $x > 0$ and concave down for $x < 0$. The change occurs at $x = 0$, which lies outside the domain, so there is no inflection point.

The inequality $e^x \geq ex$

The minimum value $e$ on the right branch says that $\dfrac{e^x}{x} \geq e$ for $x > 0$, which is the following inequality.

Equality holds only at $x = 1$. For $x \leq 0$ the left side is positive while the right side is not, so the inequality in fact holds for every real number.

The exponential integral and applications

This function has no elementary antiderivative. Instead $\int \dfrac{e^x}{x}\,dx$ is the defining integral of the special function known as the exponential integral $\operatorname{Ei}(x)$. Substituting $x = e^{u}$ turns $\int \dfrac{dx}{\ln x}$ into $\int \dfrac{e^{u}}{u}\,du$, which shows that the logarithmic integral $\operatorname{li}(x)$, the best elementary approximation to the prime counting function $\pi(x)$, is an integral of this very function. The exponential integral also arises in radiative transfer and neutron transport, wherever exponential attenuation and an inverse-distance factor act together.