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

Hint: \[x = k\] 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 \[x{\text{ }}\& {\text{ }}y \to \infty \].