Answer
398.4k+ views
Hint: In this problem, we have to simplify this expression by using limits. If we assume that f(y) is a function, then the limit of the function can be represented as: $\displaystyle \lim_{y \to c}$. This is denoted as the general expression of limit, where c is any constant value. We will use a specific formula for limit in this question:
$\Rightarrow \displaystyle \lim_{x \to a}\dfrac{{{x}^{n}}-{{a}^{n}}}{x-a}=n{{a}^{n-1}}$
Complete step by step answer:
Now, let’s solve the question.
As we know that the limit of a function f(x) is defined as a value, where the function reaches as the limit reaches some value. Limits are used for defining integration, integral calculus and continuity of the function. If we assume that f(y) is a function, then the limit of the function can be represented as;
$\displaystyle \lim_{y \to c}$
This is denoted as the general expression of limit, where c is any constant value. for this question, we will use a formula for limits and exponents:
$\Rightarrow \displaystyle \lim_{x \to a}\dfrac{{{x}^{n}}-{{a}^{n}}}{x-a}=n{{a}^{n-1}}$
Now, write the given limit:
$\Rightarrow \displaystyle \lim_{x \to 1}\dfrac{{{x}^{m}}-1}{{{x}^{n}}-1}$
Multiply its numerator and denominator such that:
$\Rightarrow \displaystyle \lim_{x \to 1}\dfrac{{{x}^{m}}-1}{x-1}\times \dfrac{x-1}{{{x}^{n}}-1}$
According to the formula $\displaystyle \lim_{x \to a}\dfrac{{{x}^{n}}-{{a}^{n}}}{x-a}=n{{a}^{n-1}}$, a = 1. So, we can write:
$\Rightarrow \displaystyle \lim_{x \to 1}\dfrac{{{x}^{m}}-{{\left( 1 \right)}^{m}}}{x-1}\times \dfrac{x-1}{{{x}^{n}}-{{\left( 1 \right)}^{n}}}$
By applying formula, we will get:
$\Rightarrow m{{.1}^{m-1}}\times \dfrac{1}{n{{.1}^{n-1}}}$
Now after simplification, we will get:
$\therefore \dfrac{m}{n}$
So, the correct answer is “Option b”.
Note: Students should note that once you apply formula for limits and exponents, the limits will be removed and then the expression will be simplified. You may notice that before applying formula, we changed the expression according to the formula to be applied because if we apply any algebraic identity, it won’t work.
$\Rightarrow \displaystyle \lim_{x \to a}\dfrac{{{x}^{n}}-{{a}^{n}}}{x-a}=n{{a}^{n-1}}$
Complete step by step answer:
Now, let’s solve the question.
As we know that the limit of a function f(x) is defined as a value, where the function reaches as the limit reaches some value. Limits are used for defining integration, integral calculus and continuity of the function. If we assume that f(y) is a function, then the limit of the function can be represented as;
$\displaystyle \lim_{y \to c}$
This is denoted as the general expression of limit, where c is any constant value. for this question, we will use a formula for limits and exponents:
$\Rightarrow \displaystyle \lim_{x \to a}\dfrac{{{x}^{n}}-{{a}^{n}}}{x-a}=n{{a}^{n-1}}$
Now, write the given limit:
$\Rightarrow \displaystyle \lim_{x \to 1}\dfrac{{{x}^{m}}-1}{{{x}^{n}}-1}$
Multiply its numerator and denominator such that:
$\Rightarrow \displaystyle \lim_{x \to 1}\dfrac{{{x}^{m}}-1}{x-1}\times \dfrac{x-1}{{{x}^{n}}-1}$
According to the formula $\displaystyle \lim_{x \to a}\dfrac{{{x}^{n}}-{{a}^{n}}}{x-a}=n{{a}^{n-1}}$, a = 1. So, we can write:
$\Rightarrow \displaystyle \lim_{x \to 1}\dfrac{{{x}^{m}}-{{\left( 1 \right)}^{m}}}{x-1}\times \dfrac{x-1}{{{x}^{n}}-{{\left( 1 \right)}^{n}}}$
By applying formula, we will get:
$\Rightarrow m{{.1}^{m-1}}\times \dfrac{1}{n{{.1}^{n-1}}}$
Now after simplification, we will get:
$\therefore \dfrac{m}{n}$
So, the correct answer is “Option b”.
Note: Students should note that once you apply formula for limits and exponents, the limits will be removed and then the expression will be simplified. You may notice that before applying formula, we changed the expression according to the formula to be applied because if we apply any algebraic identity, it won’t work.
Recently Updated Pages
Mark and label the given geoinformation on the outline class 11 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
When people say No pun intended what does that mea class 8 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Name the states which share their boundary with Indias class 9 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give an account of the Northern Plains of India class 9 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Change the following sentences into negative and interrogative class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Advantages and disadvantages of science
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Trending doubts
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Which are the Top 10 Largest Countries of the World?
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference Between Plant Cell and Animal Cell
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
10 examples of evaporation in daily life with explanations
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give 10 examples for herbs , shrubs , climbers , creepers
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Write a letter to the principal requesting him to grant class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
How do you graph the function fx 4x class 9 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)