Let $f\text{ and }g$ be two functions such that $\underset{x\to a}{\mathop{\lim }}\,f(x)$ and $\underset{x\to a}{\mathop{\lim }}\,g(x)$ exists. Then, which of the following is incomplete?
(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
360.9k+ views
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.
Last updated date: 24th Sep 2023
•
Total views: 360.9k
•
Views today: 8.60k
Recently Updated Pages
What do you mean by public facilities

Difference between hardware and software

Disadvantages of Advertising

10 Advantages and Disadvantages of Plastic

What do you mean by Endemic Species

What is the Botanical Name of Dog , Cat , Turmeric , Mushroom , Palm

Trending doubts
How do you solve x2 11x + 28 0 using the quadratic class 10 maths CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Difference Between Plant Cell and Animal Cell

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

The equation xxx + 2 is satisfied when x is equal to class 10 maths CBSE

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

Drive an expression for the electric field due to an class 12 physics CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

What is the past tense of read class 10 english CBSE
