use properties of limits to find the indicated limit. it may be necessary to rewrite an expression before…

Explanation:

Step1: Apply product - limit rule

$\lim_{x\rightarrow a}[f(x)g(x)]=\lim_{x\rightarrow a}f(x)\cdot\lim_{x\rightarrow a}g(x)$. Let $f(x)=(x + 3)^2$ and $g(x)=(4x-6)^3$. Then $\lim_{x\rightarrow2}[(x + 3)^2(4x-6)^3]=\lim_{x\rightarrow2}(x + 3)^2\cdot\lim_{x\rightarrow2}(4x-6)^3$.

Step2: Apply power - limit rule $\lim_{x\rightarrow a}[f(x)]^n=[\lim_{x\rightarrow a}f(x)]^n$

For $\lim_{x\rightarrow2}(x + 3)^2$, we have $[\lim_{x\rightarrow2}(x + 3)]^2$. Substitute $x = 2$ into $x+3$: $\lim_{x\rightarrow2}(x + 3)=2 + 3=5$, so $[\lim_{x\rightarrow2}(x + 3)]^2=5^2 = 25$.

Step3: Apply power - limit rule and sum - difference rule

For $\lim_{x\rightarrow2}(4x-6)^3$, first use the sum - difference rule $\lim_{x\rightarrow a}[f(x)-g(x)]=\lim_{x\rightarrow a}f(x)-\lim_{x\rightarrow a}g(x)$. So $\lim_{x\rightarrow2}(4x-6)=\lim_{x\rightarrow2}(4x)-\lim_{x\rightarrow2}(6)$. Then use the constant - multiple rule $\lim_{x\rightarrow a}[cf(x)]=c\lim_{x\rightarrow a}f(x)$, we get $4\lim_{x\rightarrow2}x-\lim_{x\rightarrow2}6$. Substitute $x = 2$: $4\times2-6=8 - 6=2$. Then $[\lim_{x\rightarrow2}(4x-6)]^3=2^3 = 8$.

Step4: Calculate the final limit

Multiply the two results: $\lim_{x\rightarrow2}(x + 3)^2\cdot\lim_{x\rightarrow2}(4x-6)^3=25\times8=200$.

Answer:

A. $\lim_{x\rightarrow2}[(x + 3)^2(4x-6)^3]=200$