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Question: If \(f(a) = 2,f'(a) = 1,g(a) = 1,g'(a) = 2,\) then \(\lim_{x \rightarrow a}\frac{g(x)f(a) - g(a)f(x...

If f(a)=2,f′(a)=1,g(a)=1,g′(a)=2,f(a) = 2,f'(a) = 1,g(a) = 1,g'(a) = 2, then

lim⁡x→ag(x)f(a)−g(a)f(x)x−a\lim_{x \rightarrow a}\frac{g(x)f(a) - g(a)f(x)}{x - a}equals

A

–3

B

13\frac{1}{3}

C

3

D

−13- \frac{1}{3}

Answer

3

Explanation

Solution

Applying L – Hospital's rule, we get,

lim⁡x→ag(x)f(a)−g(a)f(x)x−a=lim⁡x→ag′(x)f(a)−g(a)f′(x)1\lim_{x \rightarrow a}\frac{g(x)f(a) - g(a)f(x)}{x - a} = \lim_{x \rightarrow a}\frac{g^{'}(x)f(a) - g(a)f^{'}(x)}{1}

=g′(a)f(a)−g(a)f′(a)=2×2−1×(1)=3.= g^{'}(a)f(a) - g(a)f^{'}(a) = 2 \times 2 - 1 \times (1) = 3.