Given: The limit below
[tex]\lim_{x\to7^-}(\frac{1}{x-7})[/tex]
To Determine: The limit and the vertical asymptotes
Solution
[tex]\begin{gathered} \lim_{x\to7^-}(\frac{1}{x-7}) \\ \mathrm{For}\:x\:\mathrm{approaching}\:7\:\mathrm{from\:the\:left},\:x<7\quad \Rightarrow \quad \:x-7<0 \\ The\:denominator\:is\:a\:negative\:quantity\:approaching\:0\:from\:the\:left \\ Hence \\ \operatorname{\lim}_{x\to7^-}(\frac{1}{x-7})=-\infty \end{gathered}[/tex]
For the vertical asymptote
[tex]\begin{gathered} Vertical-asymptote \\ For\:rational\:functions,\:the\:vertical\:asymptotes\:are\:the\:undefined\:points \\ also\:known\:as\:the\:zeros\:of\:the\:denominator,\:of\:the\:simplified\:function. \end{gathered}[/tex]
The denominator of the rational function given is
[tex]\begin{gathered} denominator:x-7 \\ x-7=0 \\ x=7 \end{gathered}[/tex]
Hence:
limit = - ∞
Vertical asymptote: x = 7
