Question

If f and g are two functions such that $\underset{x\to a}{\mathop{\lim }}\,f(x)$and $\underset{x\to a}{\mathop{\lim }}\,g(x)$ exists. When $g$ is the constant function such that $g(x)=\lambda ,$ for some real number $\lambda ,$ then
(a) $\underset{x\to a}{\mathop{\lim }}\,\left[ \left( \lambda .f \right)\left( x \right) \right]=\lambda .\underset{x\to a}{\mathop{\lim }}\,f(x)$
(b)$\underset{x\to a}{\mathop{\lim }}\,\left[ \left( \dfrac{f}{\lambda } \right)(x) \right]=\dfrac{1}{\lambda }.\underset{x\to a}{\mathop{\lim }}\,f(x)$
(c) Both (a) and (b) are correct
(d) Neither (a) nor (b) is correct

Answer 1

Hint: Apply the different rules of limit (multiplication and division of two functions) and solve.

Complete step-by-step answer:
As per the given information limit of functions $f$ and $g$ exists. And $g$ is the constant function such that $g(x)=\lambda ,$ for some real number$\lambda .$
So we know constant of a limit is always a constant, therefore
$\underset{x\to a}{\mathop{\lim }}\,g(x)=\lambda ...........(i)$
Now consider the first option, simplifying the Left hand side, we get
$\underset{x\to a}{\mathop{\lim }}\,\left[ \left( \lambda .f \right)\left( x \right) \right]=\underset{x\to a}{\mathop{\lim }}\,\left[ g(x).f(x) \right]$
Now we know that the limit of the product of two functions is equal to the product of the limits. So, the limit of product of two functions is equal to the product of individual limits of the functions, therefore
$\Rightarrow \underset{x\to a}{\mathop{\lim }}\,\left[ \left( \lambda .f \right)\left( x \right) \right]=\underset{x\to a}{\mathop{\lim }}\,g(x).\underset{x\to a}{\mathop{\lim }}\,f(x)$
Substituting the value from equation (i), we get
$\Rightarrow \underset{x\to a}{\mathop{\lim }}\,\left[ \left( \lambda .f \right)\left( x \right) \right]=\lambda .\underset{x\to a}{\mathop{\lim }}\,f(x)$
So we see that the left hand side is equal to the right hand side.
Therefore, option (a) is correct.
Similarly, consider the second option, simplifying the Left hand side, we get
$\underset{x\to a}{\mathop{\lim }}\,\left[ \left( \dfrac{f}{\lambda } \right)(x) \right]=\underset{x\to a}{\mathop{\lim }}\,\left[ \dfrac{f(x)}{g(x)} \right]$
Now the limit of the quotient of two functions is equal to the quotient of the limits provided that the denominator is not equal to zero. So, the limit of quotient of two functions is equal to the quotient of individual limits of the functions with a condition and here $\lambda$ is a real number and it cannot be equal to zero, therefore the above equation can be written as,
$\underset{x\to a}{\mathop{\lim }}\,\left[ \left( \dfrac{f}{\lambda } \right)(x) \right]=\dfrac{\underset{x\to a}{\mathop{\lim }}\,f(x)}{\underset{x\to a}{\mathop{\lim }}\,g(x)}$
Substituting the value from equation (i), we get
$\underset{x\to a}{\mathop{\lim }}\,\left[ \left( \dfrac{f}{\lambda } \right)(x) \right]=\dfrac{\underset{x\to a}{\mathop{\lim }}\,f(x)}{\lambda }$
This can also be written as,
$\underset{x\to a}{\mathop{\lim }}\,\left[ \left( \dfrac{f}{\lambda } \right)(x) \right]=\dfrac{1}{\lambda }.\underset{x\to a}{\mathop{\lim }}\,f\left( x \right)$
So we see that the left hand side is equal to the right hand side.
Therefore, option (b) is also correct.
Hence the correct answer is option (c).

Note: Many students do not take notice of constant function and there are changes of getting confused. Students can also forget about the $\lambda$ as a real number and there is a possibility of selecting only option (a) as correct.