The given expression :
[tex]\text{Lim}(x\rightarrow4)\frac{3x-12}{x^2-16}[/tex]Simplify the expression:
[tex]\begin{gathered} \text{Lim}(x\rightarrow4)\frac{3x-12}{x^2-16} \\ \text{Taking 3 common from the numerator} \\ \text{Lim}(x\rightarrow4)\frac{3(x-4)}{x^2-16} \end{gathered}[/tex]Apply the square identity:
[tex]\begin{gathered} (a^2-b^2)=(a-b)(a+b) \\ \text{Lim}(x\rightarrow4)\frac{3(x-4)}{x^2-4^2} \\ \text{Lim}(x\rightarrow4)\frac{3(x-4)}{(x-4)(x+4)} \\ \text{Lim}(x\rightarrow4)\frac{3}{(x+4)} \end{gathered}[/tex]Apply the limit : x-4
[tex]\begin{gathered} \text{Lim(x}\rightarrow4)\frac{3}{x+4}=\frac{3}{4+4} \\ \text{Lim(x}\rightarrow4)\frac{3}{x+4}=\frac{3}{8} \end{gathered}[/tex]So, we get:
[tex]\text{Lim(x}\rightarrow4)\frac{3x-12}{x^2-16}=\frac{3}{8}[/tex]Answer: 3/8
