# 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 $$y=f(x)$$ at the point $$\{a,f(a)\}$$ Determine a possible function f and number a, then calculate the limit
$$\lim_{x\rightarrow2}\frac{5\cdot x^2-20}{x-2}$$

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StrycharzT

Consider the given limit $$\lim_{x\rightarrow2}\frac{5\cdot x^2-20}{x-2}$$
Note that, $$f'(a)=\lim_{x\rightarrow a}\frac{f(x)-f(a)}{x-a}$$
That implies, $$f(x)=5x^2$$ and $$a=2$$
Now obtain the limit as follows
$$\lim_{x\rightarrow2}\frac{5\cdot x^2-20}{x-2}=\lim_{x\rightarrow2}\frac{5(x^2-4)}{x-2}$$
$$=\lim_{x\rightarrow2}\frac{5(x^2-2^2)}{x-2}$$
$$=\lim_{x\rightarrow2}\frac{5(x-2)(x+2)}{(x-2)}$$
$$=\lim_{x\rightarrow2}5(x+2)$$
$$=5(2+2)$$
$$=5\cdot4$$
$$=20$$
Hence, the value of the given limit is 20.