use algebra and the properties of limits as needed to find the given limit. if the limit does not exist, say…

Explanation:

Step1: Apply limit - sum rule

$\lim_{x\rightarrow a}(f(x)+g(x))=\lim_{x\rightarrow a}f(x)+\lim_{x\rightarrow a}g(x)$. So, $\lim_{x\rightarrow 2}(2x^{4}-6x^{2}+7)=\lim_{x\rightarrow 2}(2x^{4})-\lim_{x\rightarrow 2}(6x^{2})+\lim_{x\rightarrow 2}(7)$.

Step2: Apply constant - multiple rule

$\lim_{x\rightarrow a}(cf(x)) = c\lim_{x\rightarrow a}f(x)$. Then, $\lim_{x\rightarrow 2}(2x^{4}) = 2\lim_{x\rightarrow 2}(x^{4})$, $\lim_{x\rightarrow 2}(6x^{2})=6\lim_{x\rightarrow 2}(x^{2})$.

Step3: Apply power - rule

$\lim_{x\rightarrow a}(x^{n})=a^{n}$. So, $2\lim_{x\rightarrow 2}(x^{4})=2\times2^{4}$, $6\lim_{x\rightarrow 2}(x^{2}) = 6\times2^{2}$, and $\lim_{x\rightarrow 2}(7)=7$.

Step4: Calculate each term

$2\times2^{4}=2\times16 = 32$, $6\times2^{2}=6\times4 = 24$.

Step5: Combine the terms

$32-24 + 7=15$.

Answer:

A. $\lim_{x\rightarrow 2}(2x^{4}-6x^{2}+7)=15$