Question

Which of the following statement is/are true:
(a) If \underset{x\to a}{\mathop{\lim }}\,\left\\{ f(x)+g(x) \right\\} exists, then both $\underset{x\to a}{\mathop{\lim }}\,f(x)$ and $\underset{x\to a}{\mathop{\lim }}\,g(x)$ exist.
(b) If $\underset{x\to a}{\mathop{\lim }}\,f(x)$ and $\underset{x\to a}{\mathop{\lim }}\,g(x)$ exist, then \underset{x\to a}{\mathop{\lim }}\,\left\\{ f(x)+g(x) \right\\} exists.
(c) If $\underset{x\to a}{\mathop{\lim }}\,f(x)$ and $\underset{x\to a}{\mathop{\lim }}\,g(x)$ exist, then \underset{x\to a}{\mathop{\lim }}\,\left\\{ f(x)g(x) \right\\} exists.
(d) If \underset{x\to a}{\mathop{\lim }}\,\left\\{ f(x)g(x) \right\\} exists, then both $\underset{x\to a}{\mathop{\lim }}\,f(x)$ and $\underset{x\to a}{\mathop{\lim }}\,g(x)$ exist.

Answer 1

Hint:To solve this question, one needs to check the validity of all the options using properties of the limits of functions. Apply First Principle of limit to check each of the given options or take any function which satisfies the given conditions and check if the options are correct or not.

Complete step-by-step answer:
In the given equations, we know that the limit applied to addition of two functions can be written as the sum of limits of the individual functions. Similarly, if the limit applied to the original function exists, we can apply the limit to the sum of two functions and this limit exists as well.
Thus, the statement “If \underset{x\to a}{\mathop{\lim }}\,\left\\{ f(x)+g(x) \right\\} exists, then both $\underset{x\to a}{\mathop{\lim }}\,f(x)$ and $\underset{x\to a}{\mathop{\lim }}\,g(x)$ exist” and” If $\underset{x\to a}{\mathop{\lim }}\,f(x)$ and $\underset{x\to a}{\mathop{\lim }}\,g(x)$ exist, then \underset{x\to a}{\mathop{\lim }}\,\left\\{ f(x)+g(x) \right\\} exists” are both true.
We can prove this by considering functions $f\left( x \right)=x,g\left( x \right)={{x}^{2}}$. So, we have $\underset{x\to a}{\mathop{\lim }}\,f\left( x \right)=\underset{x\to a}{\mathop{\lim }}\,(x)=a$ and $\underset{x\to a}{\mathop{\lim }}\,g\left( x \right)=\underset{x\to a}{\mathop{\lim }}\,\left( {{x}^{2}} \right)={{a}^{2}}$. Also, we have $\underset{x\to a}{\mathop{\lim }}\,f\left( x \right)+g\left( x \right)=\underset{x\to a}{\mathop{\lim }}\,\left( x \right)+\left( {{x}^{2}} \right)=a+{{a}^{2}}$.
Thus, “if \underset{x\to a}{\mathop{\lim }}\,\left\\{ f(x)+g(x) \right\\} exists, then both $\underset{x\to a}{\mathop{\lim }}\,f(x)$ and $\underset{x\to a}{\mathop{\lim }}\,g(x)$ exists” and “if $\underset{x\to a}{\mathop{\lim }}\,f(x)$ and $\underset{x\to a}{\mathop{\lim }}\,g(x)$ exist, then \underset{x\to a}{\mathop{\lim }}\,\left\\{ f(x)+g(x) \right\\} exists” holds true.
We also know that if a limit applied to two functions exists, then the limit applied to the product of functions exist as well. Thus, we have “If $\underset{x\to a}{\mathop{\lim }}\,f(x)$ and $\underset{x\to a}{\mathop{\lim }}\,g(x)$ exist, then \underset{x\to a}{\mathop{\lim }}\,\left\\{ f(x)g(x) \right\\} exists” as a correct statement.
We can prove this by considering functions $f\left( x \right)=x,g\left( x \right)={{x}^{2}}$. So, we have $\underset{x\to a}{\mathop{\lim }}\,f\left( x \right)=\underset{x\to a}{\mathop{\lim }}\,(x)=a$ and $\underset{x\to a}{\mathop{\lim }}\,g\left( x \right)=\underset{x\to a}{\mathop{\lim }}\,\left( {{x}^{2}} \right)={{a}^{2}}$. Also, we have $\underset{x\to a}{\mathop{\lim }}\,f\left( x \right)g\left( x \right)=\underset{x\to a}{\mathop{\lim }}\,\left( x \right)\left( {{x}^{2}} \right)=a\left( {{a}^{2}} \right)={{a}^{3}}$.
Thus, “If $\underset{x\to a}{\mathop{\lim }}\,f(x)$ and $\underset{x\to a}{\mathop{\lim }}\,g(x)$ exist, then \underset{x\to a}{\mathop{\lim }}\,\left\\{ f(x)g(x) \right\\} exists” holds true as well.
However, it’s not necessary that if the limit applied to a product of function exists, then the limit applied to the component functions will exist as well.
Limit of a function is a fundamental concept in calculus that analyses the behaviour of that function around a point. The notation of a limit has many applications in modern calculus. We can use the limit of a function to check differentiability of the function around any point. We can also use the limit to integrate a function defined over an interval.
When we say that a function has limit L, it means that the function gets closer and closer to the value L around the point at which the limit is applied to.
Hence, the correct answer is option (a), (b), (c).

Note: It’s necessary to know the properties of limits applied to sum and product of two or more functions. Otherwise, we won’t be able to solve this question. We can also solve this question by using the First Principle of Limit.