Question 1

Can you please help me find the limit and identify any vertical asymptotes? Thanks

Question 2

Given: The limit below

$\lim_(x\to7^-)((1)/(x-7))$

To Determine: The limit and the vertical asymptotes

Solution

$\begin{gathered} \lim_(x\to7^-)((1)/(x-7)) \\ \mathrm{For}\:x\:\mathrm{approaching}\:7\:\mathrm{from\:the\:left},\:x<7\quad \Rightarrow \quad \:x-7<0 \\ The\:denominator\:is\:a\:negative\:quantity\:approaching\:0\:from\:the\:left \\ Hence \\ \operatorname{\lim}_(x\to7^-)((1)/(x-7))=-\infty \end{gathered}$

For the vertical asymptote

$\begin{gathered} Vertical-asymptote \\ For\:rational\:functions,\:the\:vertical\:asymptotes\:are\:the\:undefined\:points \\ also\:known\:as\:the\:zeros\:of\:the\:denominator,\:of\:the\:simplified\:function. \end{gathered}$

The denominator of the rational function given is

$\begin{gathered} denominator:x-7 \\ x-7=0 \\ x=7 \end{gathered}$

Hence:

limit = - ∞

Vertical asymptote: x = 7

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