Question

If

$lim \: x \to \: a \: (f(x) + g(x)) = 6 \\ \: and \: \\ lim \: x \to \: a \: (f(x) - g(x)) = 4 \\ find \: lim \: x \to \: a \: f(x)g(x)$

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Answer 1

$\large\underline{\sf{Solution-}}$

Given that,

$\rm :\longmapsto\:\displaystyle\lim_{x \to a}[f(x) + g(x)] = 6$

can be rewritten as

$\rm :\longmapsto\:\displaystyle\lim_{x \to a}f(x) + \displaystyle\lim_{x \to a}g(x) = 6 - - - (1)$

Also, given that

$\red{\rm :\longmapsto\:\displaystyle\lim_{x \to a}[f(x) - g(x)] = 4}$

can be rewritten as

$\red{\rm :\longmapsto\:\displaystyle\lim_{x \to a}f(x) - \displaystyle\lim_{x \to a}g(x) = 4 - - - (2)}$

On adding equation (1) and equation (2), we get

$\rm :\longmapsto\:2\displaystyle\lim_{x \to a}f(x) = 10$

$\rm \implies\:\boxed{ \tt{ \: \displaystyle\lim_{x \to a}f(x) = 5 \: }} - - - (3)$

On Subtracting equation (2) from equation (1), we get

$\rm :\longmapsto\:2\displaystyle\lim_{x \to a}g(x) = 2$

$\rm \implies\:\boxed{ \tt{ \: \displaystyle\lim_{x \to a}g(x) = 1 \: }} - - - (4)$

Now, Consider

$\red{\rm :\longmapsto\:\displaystyle\lim_{x \to a}f(x)g(x)}$

$\rm \:  =  \: \displaystyle\lim_{x \to a}f(x) \: \times \: \displaystyle\lim_{x \to a}g(x)$

$\rm \:  =  \: 5 \times 1$

$\rm \:  =  \: 5$

Hence,

$\red{\rm :\longmapsto\:\boxed{ \tt{ \: \displaystyle\lim_{x \to a}f(x)g(x)} = 5 \: }}$