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_{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_{xrightarrow2}frac{5cdot x^2-20}{x-2}

Question
Limits and continuity
asked 2021-02-26
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
\(\lim_{x\rightarrow2}\frac{5\cdot x^2-20}{x-2}\)

Answers (1)

2021-02-27
Consider the given limit \(\lim_{x->2}\frac{5\cdot x^2-20}{x-2}\)
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
\(\lim_{x->2}\frac{5\cdot x^2-20}{x-2}=\lim_{x->2}\frac{5(x^2-4)}{x-2}\)
\(=\lim_{x\rightarrow2}\frac{5(x^2-2^2)}{x-2}\)
\(=\lim_{x\rightarrow2}\frac{5(x-2)(x+2)}{(x-2)}\)
\(=\lim_{x\rightarrow2}5(x+2)\)
\(=5(2+2)\)
\(=5\cdot4\)
\(=20\)
Hence, the value of the given limit is 20.
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