Find the function The following limits represent the slope of a curve y=f(x) at the point {a,f(a)} Determine a possible function f and number a, then calculate the limitlim_{xrightarrow2}frac{5cdot x^2-20}{x-2}

Find the function The following limits represent the slope of a curve $y=f\left(x\right)$ at the point $\left\{a,f\left(a\right)\right\}$ Determine a possible function f and number a, then calculate the limit
$\underset{x\to 2}{lim}\frac{5\cdot {x}^{2}-20}{x-2}$

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StrycharzT

Consider the given limit $\underset{x\to 2}{lim}\frac{5\cdot {x}^{2}-20}{x-2}$
Note that, ${f}^{\prime }\left(a\right)=\underset{x\to a}{lim}\frac{f\left(x\right)-f\left(a\right)}{x-a}$
That implies, $f\left(x\right)=5{x}^{2}$ and $a=2$
Now obtain the limit as follows
$\underset{x\to 2}{lim}\frac{5\cdot {x}^{2}-20}{x-2}=\underset{x\to 2}{lim}\frac{5\left({x}^{2}-4\right)}{x-2}$
$=\underset{x\to 2}{lim}\frac{5\left({x}^{2}-{2}^{2}\right)}{x-2}$
$=\underset{x\to 2}{lim}\frac{5\left(x-2\right)\left(x+2\right)}{\left(x-2\right)}$
$=\underset{x\to 2}{lim}5\left(x+2\right)$
$=5\left(2+2\right)$
$=5\cdot 4$
$=20$
Hence, the value of the given limit is 20.