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(a) $\underset{x\to a}{\mathop{\lim }}\,\left[ f(x)+g(x) \right]=\underset{x\to a}{\mathop{\lim }}\,f(x)+\underset{x\to a}{\mathop{\lim }}\,g(x)$

(b) $\underset{x\to a}{\mathop{\lim }}\,\left[ f(x)-g(x) \right]=\underset{x\to a}{\mathop{\lim }}\,f(x)-\underset{x\to a}{\mathop{\lim }}\,g(x)$

(c) $\underset{x\to a}{\mathop{\lim }}\,\left[ f(x).g(x) \right]=\underset{x\to a}{\mathop{\lim }}\,f(x).\underset{x\to a}{\mathop{\lim }}\,g(x)$

(d) $\underset{x\to a}{\mathop{\lim }}\,\left[ \dfrac{f(x)}{g(x)} \right]=\dfrac{\underset{x\to a}{\mathop{\lim }}\,f(x)}{\underset{x\to a}{\mathop{\lim }}\,g(x)}$

Answer
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Hint: Basic properties of limit is required to solve this kind of problem

Complete step-by-step answer:

As per the given information the functions $\underset{x\to a}{\mathop{\lim }}\,f(x)\text{ and }\underset{x\to a}{\mathop{\lim }}\,g(x)$exists.

Now we will consider the options separately.

We know the limit of sum is the sum of limits. So, the limit of sum of two functions is equal to the sum of individual limits of the functions, that is,

$\underset{x\to a}{\mathop{\lim }}\,\left[ f(x)+g(x) \right]=\underset{x\to a}{\mathop{\lim }}\,f(x)+\underset{x\to a}{\mathop{\lim }}\,g(x)$

This option is correct.

We also know the limit of difference is the difference of the limits. So, the limit of difference of two functions is equal to the difference of individual limits of the functions, that is,

$\underset{x\to a}{\mathop{\lim }}\,\left[ f(x)-g(x) \right]=\underset{x\to a}{\mathop{\lim }}\,f(x)-\underset{x\to a}{\mathop{\lim }}\,g(x)$

So, this option is also correct.

Now we know the limit of a product is the product of the limits. So, the limit of product of two functions is equal to the product of individual limits of the functions, that is,

$\underset{x\to a}{\mathop{\lim }}\,\left[ f(x).g(x) \right]=\underset{x\to a}{\mathop{\lim }}\,f(x).\underset{x\to a}{\mathop{\lim }}\,g(x)$

So, this option is also correct.

In functions, the law governing the limits clearly states that the limit value is changed for division of two variable functions.

Now the limit of quotients is 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 that the $g(x)\ne 0$, that is,

$\underset{x\to a}{\mathop{\lim }}\,\left[ \dfrac{f(x)}{g(x)} \right]=\dfrac{\underset{x\to a}{\mathop{\lim }}\,f(x)}{\underset{x\to a}{\mathop{\lim }}\,g(x)}$, such that $\underset{x\to a}{\mathop{\lim }}\,g(x)\ne 0$

But this condition is not provided in the question. So we don’t know whether the denominator is zero or not.

Hence this option is incomplete.

So, this option is not correct.

Thus, the correct answer is option (d).

Note: Students make mistakes by just observing the equations. Then forget to notice that the condition $\underset{x\to a}{\mathop{\lim }}\,g(x)\ne 0$ is not given. They think all the options are correct. So be careful.

Complete step-by-step answer:

As per the given information the functions $\underset{x\to a}{\mathop{\lim }}\,f(x)\text{ and }\underset{x\to a}{\mathop{\lim }}\,g(x)$exists.

Now we will consider the options separately.

We know the limit of sum is the sum of limits. So, the limit of sum of two functions is equal to the sum of individual limits of the functions, that is,

$\underset{x\to a}{\mathop{\lim }}\,\left[ f(x)+g(x) \right]=\underset{x\to a}{\mathop{\lim }}\,f(x)+\underset{x\to a}{\mathop{\lim }}\,g(x)$

This option is correct.

We also know the limit of difference is the difference of the limits. So, the limit of difference of two functions is equal to the difference of individual limits of the functions, that is,

$\underset{x\to a}{\mathop{\lim }}\,\left[ f(x)-g(x) \right]=\underset{x\to a}{\mathop{\lim }}\,f(x)-\underset{x\to a}{\mathop{\lim }}\,g(x)$

So, this option is also correct.

Now we know the limit of a product is the product of the limits. So, the limit of product of two functions is equal to the product of individual limits of the functions, that is,

$\underset{x\to a}{\mathop{\lim }}\,\left[ f(x).g(x) \right]=\underset{x\to a}{\mathop{\lim }}\,f(x).\underset{x\to a}{\mathop{\lim }}\,g(x)$

So, this option is also correct.

In functions, the law governing the limits clearly states that the limit value is changed for division of two variable functions.

Now the limit of quotients is 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 that the $g(x)\ne 0$, that is,

$\underset{x\to a}{\mathop{\lim }}\,\left[ \dfrac{f(x)}{g(x)} \right]=\dfrac{\underset{x\to a}{\mathop{\lim }}\,f(x)}{\underset{x\to a}{\mathop{\lim }}\,g(x)}$, such that $\underset{x\to a}{\mathop{\lim }}\,g(x)\ne 0$

But this condition is not provided in the question. So we don’t know whether the denominator is zero or not.

Hence this option is incomplete.

So, this option is not correct.

Thus, the correct answer is option (d).

Note: Students make mistakes by just observing the equations. Then forget to notice that the condition $\underset{x\to a}{\mathop{\lim }}\,g(x)\ne 0$ is not given. They think all the options are correct. So be careful.

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