Answer :

Certainly! Let's find the limit of the function [tex]\( g(x) = x^3 \)[/tex] as [tex]\( x \)[/tex] approaches 2 for the given expression:

[tex]\[ \lim _{x \rightarrow 2} \frac{g(x)-g(2)}{x-2} \][/tex]

First, let's evaluate the function [tex]\( g(x) \)[/tex] at [tex]\( x = 2 \)[/tex]:

[tex]\[ g(2) = 2^3 = 8 \][/tex]

Now, substitute [tex]\( g(x) = x^3 \)[/tex] and [tex]\( g(2) = 8 \)[/tex] into the limit expression:

[tex]\[ \lim _{x \rightarrow 2} \frac{x^3 - 8}{x - 2} \][/tex]

Next, we need to simplify the expression inside the limit. Notice that [tex]\( x^3 - 8 \)[/tex] is a difference of cubes, which can be factored as follows:

[tex]\[ x^3 - 8 = (x - 2)(x^2 + 2x + 4) \][/tex]

So, the limit expression becomes:

[tex]\[ \lim _{x \rightarrow 2} \frac{(x - 2)(x^2 + 2x + 4)}{x - 2} \][/tex]

Since [tex]\( x \neq 2 \)[/tex] in the limit (we are approaching 2 but not equal to 2), we can cancel out [tex]\( (x - 2) \)[/tex] in the numerator and the denominator:

[tex]\[ \lim _{x \rightarrow 2} (x^2 + 2x + 4) \][/tex]

Now, we simply evaluate the remaining polynomial [tex]\( x^2 + 2x + 4 \)[/tex] at [tex]\( x = 2 \)[/tex]:

[tex]\[ 2^2 + 2(2) + 4 = 4 + 4 + 4 = 12 \][/tex]

Thus, the limit is:

[tex]\[ \lim _{x \rightarrow 2} \frac{g(x) - g(2)}{x - 2} = 12 \][/tex]

So, the limit of the given expression is [tex]\( 12 \)[/tex]. Additionally, we found that:

[tex]\[ g(2) = 8 \][/tex]

Putting these together:

[tex]\[ (g(2), \lim _{x \rightarrow 2} \frac{g(x) - g(2)}{x - 2}) = (8, 12) \][/tex]

Therefore, the limit and the value of the function at [tex]\( x = 2 \)[/tex] are 8 and 12, respectively.