Answer:
The given function is :
[tex]\lim_{x\rightarrow 7}\frac{1}{(x-7)^2}[/tex]
For finding the required limit : We use the following property of limits
[tex]\lim_{x\rightarrow a^-}f(x)=L,\lim_{x\rightarrow a^+}f(x)=L\implies \lim_ {x\rightarrow a}f(x)=L[/tex]
Now, if approaching from left (x-7) < 0 but its square is positive So the whole limit becomes +∞
If approaching from right (x - 7)_> 0 and its square also positive so the whole limit equals +∞
Hence, the required limit is +∞
Now, to find the vertical asymptote : check the value at which the function is not defined.
At x = 7 the denominator becomes 0 and the function is not defined so x = 7 is the vertical asymptote of the given function.
