Question

Find the vertical and horizontal asymptote of the curve $y = \dfrac{{{e^x}}}{x}$ using the definition of both horizontal and vertical asymptote.

Answer 1

$x = k$ is asymptote to the curve $y = f(x)$if
$\mathop {\lim }\limits_{x \to {k^ + }} f(x) = + \infty or - \infty \& \mathop {\lim }\limits_{x \to {k^ - }} f(x) = + \infty or - \infty$
So by putting the limit we will find the vertical asymptote.
Horizontal asymptotes correspond to the range of a function.

Complete step-by-step answer:
It is given that, $y = \dfrac{{{e^x}}}{x}$
We need to find out the vertical & horizontal asymptotes to the curve $y = \dfrac{{{e^x}}}{x}$
Vertical asymptote:
We know that x=k is asymptote to the curve $y = f(x)$if
$\mathop {\lim }\limits_{x \to {k^ + }} f(x) = + \infty {\rm{ or }} - \infty \& \mathop {\lim }\limits_{x \to {k^ - }} f(x) = + \infty {\rm{ or }} - \infty$
That is vertical asymptotes are found when the function is not defined. That is the denominator must be 0 for this to occur here since $y = \dfrac{{{e^x}}}{x}$ we say that $f(x) = \dfrac{{{e^x}}}{x}$
$x = 0$ is a asymptote to the curve $y = \dfrac{{{e^x}}}{x}$if
$\mathop {\lim }\limits_{x \to {0^ + }} f(x) = + \infty {\rm{ or}} - \infty \& \mathop {\lim }\limits_{x \to {0^ - }} f(x) = + \infty {\rm{ or}} - \infty$
Now, let us substitute the value for $f(x)$in both the above limits we get,$\mathop {\lim }\limits_{x \to {0^ + }} \dfrac{{{e^x}}}{x} = + \infty \& \mathop {\lim }\limits_{x \to {0^ - }} \dfrac{{{e^x}}}{x} = + \infty$
Hence at $x=0$, there is a vertical asymptote.
Horizontal asymptote:
We know $y = k$ is asymptote to the curve $y = f(x)$ if
$\mathop {\lim }\limits_{x \to {\infty ^ + }} y = k\& \mathop {\lim }\limits_{x \to {\infty ^ - }} y = k$
Horizontal asymptotes correspond to the range of a function.
Here$y$ is defined for all values of $x$.
However, any values of $x,y$ can never be 0.
This is because ${e^x}$ must be zero for a certain value of x.
However, as this is not possible, there exists an asymptote at $\;y = 0$.
Hence at $\;y = 0$ there is a horizontal asymptote.

Hence the horizontal and vertical asymptotes for the curve $y = \dfrac{{{e^x}}}{x}$ is $x = 0$ and $\;y = 0$

Note: A straight line is said to be an asymptote to the curve $y = f\left( x \right)$ if the distance of point $P\left( {x,y} \right)$ on the curve from the line tends to zero when $x \to \infty$ or $y \to \infty$ both $x{\text{ }}\& {\text{ }}y \to \infty$.