Limits and asymptotes of a homographic function

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Consider the function $f(x) = \frac{1-2x}{x-3}.$

• (a) Find $\displaystyle\lim_{x\to 3^-} f(x)$ and $\displaystyle\lim_{x\to +\infty} f(x)$.
• (b) State the equations of the vertical and horizontal asymptotes.

Solution

(a) As $x\to 3^-$: the numerator tends to $1-6=-5$, the denominator $x-3\to 0^-$. Hence $\lim_{x\to 3^-}\frac{1-2x}{x-3} = \frac{-5}{0^-} = +\infty.$ As $x\to+\infty$ the leading terms dominate: $\dfrac{-2x}{x}\to -2$, so $\lim_{x\to+\infty} f(x) = -2.$

(b) The denominator vanishes at $x=3$ (where the function diverges): vertical asymptote $x=3$. The finite limit $-2$ at infinity gives the horizontal asymptote $y=-2$. $\boxed{\lim_{x\to 3^-}f=+\infty;\quad \lim_{x\to+\infty}f=-2;\quad \text{vert. as. } x=3;\quad \text{horiz. as. } y=-2}$