Let P(x)=a1 x+a2 x^2+a3 x^3+…….+a100 x^100, where a1= 1 and ai ∈R ∀i=2,3,4, …, 100 then limx →0

Name: Problem 49, Chapter 1: Differential Calculus for JEE Main and Advanced
Uploaded: 2022-02-13T07:46:58-08:00
Duration: 2 min 5 s
Channel: Sai Sai
Description: Let P(x)=a1 x+a2 x^2+a3 x^3+…….+a100 x^100, where a1= 1 and ai ∈R ∀i=2,3,4, …, 100 then limx →0 (√(1+P(x))-1)/(x) has the value equal to (A) 100 (B) (1)/(100) (C) 1 (D) 5050

step 1 Hello everyone from the given question we can return it as limit extends to 0 limit of 1 plus p

Let P(x)=a1 x+a2 x^2+a3 x^3+…….+a100 x^100, where a1= 1 and...

Let $\mathrm{P}(\mathrm{x})=\mathrm{a}_{1} \mathrm{x}+\mathrm{a}_{2} \mathrm{x}^{2}+\mathrm{a}_{3} \mathrm{x}^{3}+\ldots \ldots .+\mathrm{a}_{100} \mathrm{x}^{100}$, where $\mathrm{a}_{1}=$ 1 and $a_{i} \in R \forall i=2,3,4, \ldots, 100$ then $\lim _{x \rightarrow 0} \frac{\sqrt[100]{1+P(x)}-1}{x}$ has the value equal to (A) 100 (B) $\frac{1}{100}$ (C) 1 (D) 5050

Let $\mathrm{P}(\mathrm{x})=\mathrm{a}_{1} \mathrm{x}+\mathrm{a}_{2} \mathrm{x}^{2}+\mathrm{a}_{3} \mathrm{x}^{3}+\ldots \ldots .+\mathrm{a}_{100} \mathrm{x}^{100}$, where $\mathrm{a}_{1}=$
1 and $a_{i} \in R \forall i=2,3,4, \ldots, 100$
then $\lim _{x \rightarrow 0} \frac{\sqrt[100]{1+P(x)}-1}{x}$ has the value equal to
(A) 100
(B) $\frac{1}{100}$
(C) 1
(D) 5050

Key Concepts

Limits

Limits deal with the behavior of a function as its input approaches a particular point. They are fundamental in calculus, especially in determining the instantaneous rate of change of a function at a point. This concept is crucial for understanding continuity, differentiation, and the definition of the derivative.

Taylor Series Expansion

The Taylor series expansion is a method to represent a smooth function as an infinite sum of terms calculated from the derivatives of the function at a single point. In problems involving limits or approximations near a point, especially when the input value is small, Taylor series allows for the approximation of non-linear functions by their linear terms.

Binomial Approximation

The binomial approximation is a specific case of the Taylor series expansion used for expressions of the form (1 + u)^n when u is small. It simplifies complex expressions by expanding them into a series, usually truncating after the first-order term, which is particularly useful in limit problems to determine the leading behavior of the function.

Asymptotic Analysis

Asymptotic analysis involves studying the behavior of functions as the variables approach a limiting value, often zero or infinity. By identifying the dominant or leading order term, one can approximate complex expressions and effectively evaluate limits, making it a key tool in analyzing problems that involve series expansions and approximations.

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