First of all, you have to understand [tex]g[/tex] is a square-root function.
Square-root functions are continuous across their entire domain, and their domain is all real x-values for which the expression within the square-root is non-negative.
In other words, for any square-root function [tex]q[/tex] and any input [tex]c[/tex] in the domain of [tex]q[/tex] (except for its endpoint), we know that this equality holds: [tex]lim \ q(x)=q(c)[/tex]
Let's take [tex] \sqrt{x} [/tex] as an example.
The domain of [tex]\sqrt x[/tex] is all real numbers such that [tex]x \geq 0[/tex]. Since [tex]x=0[/tex] is the endpoint of the domain, the two-sided limit at that point doesn't exist (you can't approach [tex]0[/tex] from the left).
However, continuity at an endpoint only demands that the one-sided limit is equal to the function's value:
[tex]lim \ \sqrt{x} = \sqrt{0} =0[/tex]
In conclusion, the equality [tex]lim \ q(x)=q(c)[/tex] holds for any square-root function [tex]q[/tex] and any real number [tex]c[/tex] in the domain of [tex]q[/tex] except for its endpoint, where the two-sided limit should be replaced with a one-sided limit.
The input [tex]x=-3[/tex], is within the domain of [tex]g[/tex].
Therefore, in order to find [tex]lim \ g(x) [/tex] we can simply evaluate [tex]g[/tex] at [tex]x-3[/tex].
[tex]g(x) [/tex]
[tex] \sqrt{7x+22} [/tex]
[tex] \sqrt{7(-3)+22} [/tex]
[tex] \sqrt{1} =1[/tex]
