Maths-
General
Easy

Question

lim for x not stretchy rightwards arrow 0 of   fraction numerator tan invisible function application x minus sin invisible function application x over denominator x cubed end fraction is equal to

  1. 1 half
  2. negative 1 half
  3. 0
  4. 1

hintHint:

A limit is a value that a function approaches as input and yields some value, according to the definition of limits in mathematics. Calculus and mathematical analysis depend on limits, which are also used to determine integrals, derivatives, and continuity. Here we have given lim for x not stretchy rightwards arrow 0 of   fraction numerator tan invisible function application x minus sin invisible function application x over denominator x cubed end fraction and we have to find the solution to this. 

The correct answer is: 1 half


    Now we know that Limit describes the value that a function or sequence approaches when its input approaches a certain value. A connection between derivative and integral that causes them to both be defined by the idea of limit. Additionally, they are inverse operations of one another. The fundamental theorem of calculus is the name given to this property. Additionally, each of these is essential to contemporary science.
    The area under the curve of the mathematical function f(x), which is plotted as a function of x, is referred to as the integral of a function. The slope of the curve for the mathematical function f(x), which is shown as a function of x, is referred to as a function's derivative.
    Here we have given:
    lim for x not stretchy rightwards arrow 0 of   fraction numerator tan invisible function application x minus sin invisible function application x over denominator x cubed end fraction
W e space c a n space w r i t e space i t space a s colon
lim for x not stretchy rightwards arrow 0 of   fraction numerator begin display style fraction numerator sin x over denominator cos x end fraction end style minus sin x over denominator x cubed end fraction
S i m p l i f y i n g space i t comma space w e space g e t colon
lim for x not stretchy rightwards arrow 0 of   fraction numerator begin display style sin x minus cos x end style over denominator cos x cross times x cubed end fraction
lim for x not stretchy rightwards arrow 0 of   fraction numerator begin display style sin x left parenthesis 1 minus cos x right parenthesis end style over denominator cos x cross times x cubed end fraction
lim for x not stretchy rightwards arrow 0 of   fraction numerator tan x over denominator x end fraction cross times fraction numerator begin display style 2 cross times sin squared x over 2 end style over denominator x squared end fraction
lim for x not stretchy rightwards arrow 0 of   fraction numerator tan x over denominator x end fraction cross times stack space lim with space x not stretchy rightwards arrow 0 below space fraction numerator begin display style 2 cross times sin squared x over 2 end style over denominator x squared end fraction
N o w space m u l t i p l y i n g space a n d space d i v i d i n g space b y space 4 space i n space x squared comma space w e space g e t colon
lim for x not stretchy rightwards arrow 0 of   fraction numerator tan x over denominator x end fraction cross times stack space lim with space x not stretchy rightwards arrow 0 below space fraction numerator begin display style 2 cross times sin squared x over 2 end style over denominator begin display style fraction numerator 4 cross times x squared over denominator 4 end fraction end style end fraction
A p p l y i n g space t h e space l i m i t s comma space w e space g e t colon
1 cross times open parentheses fraction numerator sin begin display style x over 2 end style over denominator begin display style x over 2 end style end fraction close parentheses squared equals 1 cross times 1 half cross times 1 equals 1 half

    Calculus and mathematical analysis depend on limits, which are also used to determine integrals, derivatives, and continuity. A function with a value that approaches the input is said to have a limit. So the answer is 1/2 for the given expression.

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