Question

question
find the equation of all vertical asymptotes of the following function.
f(x)=\frac{2 x-3}{sqrt{x^{2}-3 x-28}}
answer attempt 1 out of 2

Explanation:

Step1: Find the domain of the function

For the function \(f(x)=\frac{2x - 3}{\sqrt{x^{2}-3x - 28}}\), the expression under the square - root must be positive, i.e., \(x^{2}-3x - 28>0\).
Factor the quadratic: \(x^{2}-3x - 28=(x - 7)(x+4)>0\).
The solutions of the inequality \((x - 7)(x + 4)>0\) are \(x<-4\) or \(x>7\) using the sign - chart method (test intervals \((-\infty,-4)\), \((-4,7)\) and \((7,\infty)\)).

Step2: Analyze the behavior near the boundary points

Vertical asymptotes occur where the function approaches \(\pm\infty\).
For a rational function \(y=\frac{N(x)}{D(x)}\) (here \(N(x)=2x - 3\) and \(D(x)=\sqrt{x^{2}-3x - 28}\)), we consider the limit as \(x\) approaches the boundary points of the domain.
\(\lim_{x
ightarrow - 4^{-}}\frac{2x - 3}{\sqrt{x^{2}-3x - 28}}\), let \(t=x + 4\), \(x=t - 4\). Then \(x^{2}-3x - 28=(t - 4)^{2}-3(t - 4)-28=t^{2}-8t + 16-3t + 12-28=t^{2}-11t\). As \(t
ightarrow0^{-}\), \(\sqrt{x^{2}-3x - 28}=\sqrt{t(t - 11)}\sim\sqrt{- 11t}\) (for \(t
ightarrow0^{-}\)) and \(2x-3=2(t - 4)-3=2t-11\sim - 11\). So \(\lim_{x
ightarrow - 4^{-}}\frac{2x - 3}{\sqrt{x^{2}-3x - 28}}=\lim_{t
ightarrow0^{-}}\frac{2t-11}{\sqrt{t(t - 11)}}\) is a finite non - zero value (since the numerator approaches \(-11\) and the denominator approaches \(0\) from the positive side as \(x
ightarrow - 4^{-}\), but the function is not defined on an open interval around \(x = - 4\) (the domain is \(x<-4\) or \(x>7\)).
Similarly, for \(\lim_{x
ightarrow7^{+}}\frac{2x - 3}{\sqrt{x^{2}-3x - 28}}\), let \(u=x - 7\), \(x=u + 7\). Then \(x^{2}-3x - 28=(u + 7)^{2}-3(u + 7)-28=u^{2}+14u+49-3u - 21-28=u^{2}+11u\). As \(u
ightarrow0^{+}\), \(\sqrt{x^{2}-3x - 28}=\sqrt{u(u + 11)}\sim\sqrt{11u}\) and \(2x-3=2(u + 7)-3=2u + 11\sim11\). The function is not defined on an open interval around \(x = 7\) (the domain is \(x<-4\) or \(x>7\)).

Answer:

No Vertical Asymptotes