Eslatib o'tamiz, \(x = a\) da vertikal asimptotlar paydo bo'ladi, agar \(x = a\) dagi chegaralardan birortasi (bir tomonlama yoki umumiy chegara) ortiqcha yoki minus cheksizlik bo'lsa.

Oldingi misollardan ko'rishimiz mumkinki, ratsional ifodalar uchun nolga bo'linish mavjud bo'lganda vertikal asimptotlar paydo bo'ladi. Shuning uchun, yagona mumkin bo'lgan vertikal asimptota \(x={\textstyle{{10} \over 3}}\) da bo'ladi.

Keling, ikkita bir tomonlama chegaralarni baholab, bu aslida vertikal asimptota ekanligini tekshirib ko'raylik,

\[\mathop {\lim }\limits_{x \to \,{{{\textstyle{{10} \over 3}}}^{\, - }}} \frac{{7x}}{{{{\left( {10 - 3x} \right)}^4}}}\hspace{0.25in}\,\,\,\,{\rm{and}}\hspace{0.25in}\hspace{0.25in}\mathop {\lim }\limits_{x \to \,{{{\textstyle{{10} \over 3}}}^{\, + }}} \frac{{7x}}{{{{\left( {10 - 3x} \right)}^4}}}\]

Ikkala holatda ham \(x \to {\textstyle{{10} \over 3}}\) (chapdan ham, o'ngdan ham) surat \({\textstyle{{70} \over 3}}\) ga o'tadi.

Bir tomonlama limitlar uchun bizda quyidagi ma'lumotlar mavjud:

\[\begin{align*}x \to {{\textstyle{{10} \over 3}}^ - }\hspace{0.25in} \Rightarrow \hspace{0.25in}x < {\textstyle{{10} \over 3}}\hspace{0.25in}\, \Rightarrow \hspace{0.25in}{\textstyle{{10} \over 3}} - x > 0\hspace{0.25in}\, \Rightarrow \hspace{0.25in}10 - 3x > 0\\ x \to {{\textstyle{{10} \over 3}}^ + }\hspace{0.25in} \Rightarrow \hspace{0.25in}x > {\textstyle{{10} \over 3}}\hspace{0.25in}\, \Rightarrow \hspace{0.25in}{\textstyle{{10} \over 3}} - x < 0\hspace{0.25in}\, \Rightarrow \hspace{0.25in}10 - 3x < 0\end{align*}\]

Endi, maxrajdagi ko'rsatkich tufayli biz bir tomonlama chegaralarning har biri uchun bizda bo'lishini ko'rishimiz mumkin,

\[{\left( {10 - 3x} \right)^4} \to {0^ + }\]

Demak, har ikki holatda ham chegarada surat belgilangan musbat songa yaqinlashadi va maxraj musbat va borgan sari kichik bo‘ladi. Shunday qilib, biz ega bo'lamiz,

\[\mathop {\lim }\limits_{x \to \,{{{\textstyle{{10} \over 3}}}^{\, - }}} \frac{{7x}}{{{{\left( {10 - 3x} \right)}^4}}} = \infty \hspace{0.25in}\hspace{0.25in}\,\mathop {\lim }\limits_{x \to \,{{{\textstyle{{10} \over 3}}}^{\, + }}} \frac{{7x}}{{{{\left( {10 - 3x} \right)}^4}}} = \infty \hspace{0.25in}\,\,\,\,\,\mathop {\lim }\limits_{x \to \,{\textstyle{{10} \over 3}}} \frac{{7x}}{{{{\left( {10 - 3x} \right)}^4}}} = \infty \]

ushbu limitlarning har biri \(x = {\textstyle{{10} \over 3}}\) da vertikal asimptota mavjudligini ko'rsatadi.