Question:

For x \in R, limx(x3x+2)x lim_{ x \to \infty} \bigg( \frac{x - 3}{ x + 2}\bigg)^x is equal to

Updated On: Jul 14, 2022
  • e
  • e1 e^{-1}
  • e5e^{-5}
  • e5e^5
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The Correct Option is D

Solution and Explanation

For x \in R, limx(x3x+2)x=limx(13/x)x(1+2/x)=e3e2=e5 lim_{ x \to \infty} \bigg( \frac{x - 3}{ x + 2}\bigg)^x = lim_{x \to \infty} \frac{(1 - 3 / x)^x }{ (1 + 2 / x)} = \frac{e^{-3}}{e^2} = e^{-5}
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Concepts Used:

Relations

A relation in mathematics defines the relationship between two different sets of information. If two sets are considered, the relation between them will be established if there is a connection between the elements of two or more non-empty sets. Therefore, we can say, ‘A set of ordered pairs is defined as a relation.’

Read Also: Relation and Function

Types of Relations:

There are 8 main types of relations which are:

  1. Empty Relation - An empty relation is one in which there is no relation between any elements of a set.
  2. Universal Relation - A universal is a type of relation in which every element of a set is related to each other. Now one of the universal relations will be R = {x, y} where, |x – y| ≥ 0. For universal relation, R = A × A
  3. Identity Relation - In an identity relation, every element of a set is related to itself only. For example, in a set A = {a, b, c}, the identity relation will be I = {a, a}, {b, b}, {c, c}.
  4. Inverse Relation - It is seen when a set has elements which are inverse pairs of another set. For example if set A = {(a, b), (c, d)}, then inverse relation will be R-1 = {(b, a), (d, c)}.
  5. Reflexive Relation - If every element of set A maps for itself, then set A is known as a reflexive relation.It is represented as a∈ A, (a,a) ∈ R.
  6. Symmetric Relation - A relation R on a set A is known as asymmetric relation if (a, b) ∈R then (b, a) ∈R , such that for all a and b ∈A.
  7. Transitive Relation - For transitive relation, if (x, y) ∈ R, (y, z) ∈ R, then (x, z) ∈ R. For a transitive relation, aRb and bRc ⇒ aRc ∀ a, b, c ∈ A
  8. Equivalence Relation - If a relation is reflexive, symmetric and transitive at the same time it is known as an equivalence relation.

Representation of Relations:

There are two ways by which a relation can be represented-

  1. Roster method
  2. Set-builder method

The roster form and set-builder for for a set integers lying between -2 and 3 will be-

Roster form

I= {-1,0,1,2}

Set-builder form

I= {x:x∈I,-2<x<3}