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}

Question

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}

EunoR 2021-02-26 Answered

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
$lim_{x \to 2} \frac{5 \cdot x^{2} - 20}{x - 2}$

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Expert Answer

StrycharzT

Answered 2021-02-27 Author has 102 answers

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

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