Question

Find the rectilinear asymptotes to the curve
Y=xe^1/x​

Answer 1

Find the rectilinear asymptotes to the curve

Y=xe^1/x​

Answer:

line x = 0 is vertical asymptote

and

y = x + 1 is oblique asymptote.

Step-by-step explanation:

Since

y = x e^(1/x)

Taking the right hand sided limit as x approaching to zero

we get

$\lim_{x \to \ 0} xe^\frac{1}{x}\\= \lim_{x \to \ 0} \frac{e^\frac{1}{x}}{1/x} \\= \lim_{x \to \ 0} \frac{e^\frac{1}{x}}{1/x} \\ L-Hipotial rule \\= \lim_{x \to \ 0} \frac{e^\frac{1}{x}(-1/x^2)}{-1/x^2)} \\= \lim_{x \to \ 0} e^\frac{1}{x}\\= infinity\\\\Thus$

line x = 0 is vertical asymptote

The horizontal asymptote do not exist instead their exist a oblique asymptote which is

line y = x + 1

See below graph for details .

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