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Gujarati JEE Mathematics : લક્ષ-સાતત્ય અને વિકલન

Multiple Choice Questions

શુન્યેત્તર સતત વિધેય f એ f(x + y) = f(x) f(y0, ∀ x, y ∈R નું પાલન કરે છે. જો f(2) = 9 તો f(3) = .......

જો $bold lim with bold x bold rightwards arrow bold 0 below bold space fraction numerator bold log begin display style bold space end style begin display style bold left parenthesis end style begin display style bold space end style begin display style bold 3 end style begin display style bold space end style begin display style bold plus end style begin display style bold space end style begin display style bold x end style begin display style bold right parenthesis end style begin display style bold space end style begin display style bold minus end style begin display style bold space end style begin display style bold log end style begin display style bold left parenthesis end style begin display style bold 3 end style begin display style bold space end style begin display style bold minus end style begin display style bold space end style begin display style bold x end style begin display style bold right parenthesis end style over denominator bold x end fraction bold space bold equals bold space bold k bold space bold ત ો bold space bold k bold space bold equals bold space bold. bold. bold. bold. bold. bold. bold. bold. bold space$

0
$fraction numerator bold minus bold 1 over denominator bold 3 end fraction$
$fraction numerator bold minus bold 2 over denominator bold 3 end fraction$
$bold 2 over bold 3$

$bold f bold left parenthesis bold x bold right parenthesis bold space bold equals bold space open curly brackets table attributes columnalign center end attributes row cell fraction numerator bold log bold left parenthesis bold 1 bold plus bold x bold plus bold x to the power of bold 2 bold right parenthesis bold space bold plus bold space bold log bold left parenthesis bold 1 bold minus bold x bold plus bold x to the power of bold 2 bold right parenthesis over denominator bold sec bold space bold x bold space bold minus bold space bold cosx end fraction bold semicolon bold x bold # bold 0 end cell row cell bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold k bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold semicolon bold space bold x bold equals bold space bold 0 bold space end cell end table close$ જો વિધેય f એ x = 0 આગળ સતત હોય, તો k = ..........

0
-1
1
2

bold n bold space bold element of bold space bold N

ની કઈ કિંમત માટે $bold lim with bold x bold rightwards arrow bold 0 below bold space bold equals bold space fraction numerator bold left parenthesis bold cosx bold space bold minus bold space bold 1 bold right parenthesis bold space bold left parenthesis bold cos bold space bold x bold space bold minus bold space bold e to the power of bold x bold right parenthesis over denominator bold x to the power of bold n end fraction$ સાન્ત શુન્યેત્તર સંખ્યા મળે ?

Tips: -

(cos x - 1) (cos x - e^x)

$bold equals bold space open parentheses open parentheses bold 1 bold minus fraction numerator bold x to the power of bold 2 over denominator bold 2 bold factorial end fraction bold space bold plus fraction numerator bold x to the power of bold 4 over denominator bold 4 bold factorial end fraction bold. bold. bold. close parentheses bold minus bold 1 close parentheses bold space open square brackets open parentheses bold 1 bold minus bold space fraction numerator bold x to the power of bold 2 over denominator bold 2 bold factorial end fraction bold plus fraction numerator bold x to the power of bold 4 over denominator bold 4 bold factorial end fraction bold. bold. bold. close parentheses bold minus bold space open parentheses bold 1 bold plus bold x bold plus fraction numerator bold x to the power of bold 2 over denominator bold 2 bold factorial end fraction bold plus fraction numerator bold x to the power of bold 3 over denominator bold 3 bold factorial end fraction bold plus bold space bold. bold. bold. bold. close parentheses close square brackets bold equals bold space bold x to the power of bold 2 bold space open parentheses fraction numerator bold minus bold 1 over denominator bold 2 end fraction bold plus bold space fraction numerator bold x to the power of bold 2 over denominator bold 4 bold factorial end fraction bold minus bold. bold. bold. bold. bold. close parentheses bold space open parentheses bold minus bold 1 bold space bold minus bold space bold x to the power of bold 2 bold space bold minus bold space fraction numerator bold x to the power of bold 3 over denominator bold 3 bold factorial end fraction bold minus bold space bold. bold. bold. close parentheses bold equals bold space bold x to the power of bold 3 bold space open parentheses fraction numerator bold minus bold 1 over denominator bold 2 end fraction bold plus bold space fraction numerator bold x to the power of bold 2 over denominator bold 4 bold factorial end fraction bold minus bold. bold. bold. bold. bold. close parentheses open parentheses bold minus bold 1 bold space bold minus bold space bold x bold minus bold space fraction numerator bold x to the power of bold 3 over denominator bold 3 bold factorial end fraction bold minus bold space bold. bold. bold. close parentheses bold equals bold space bold lim with bold x bold rightwards arrow bold 0 below bold space fraction numerator bold left parenthesis bold cos bold space bold x bold space bold minus bold space bold 1 bold right parenthesis bold space bold left parenthesis bold cos bold space bold x bold space bold minus bold space bold e to the power of bold x bold right parenthesis over denominator bold x to the power of bold 3 end fraction bold space bold equals bold space bold 1 over bold 2 bold space bold # bold space bold 0$

∴ n = 3

નોંધ : 1, 2 માટે લક્ષ શૂન્ય થાય તથા n > 3 માટે સાન્ત લક્ષ ના મળે.

વિધેય f સતત છે. જો f(x) f $open parentheses bold 1 over bold x close parentheses$ = f(x) + f $open parentheses bold 1 over bold x close parentheses bold comma bold space bold for all$ x ∈ Df અને f(1) > તો $bold lim with bold x bold rightwards arrow bold 1 below$ f(x) = .......

1
2
3
લક્ષ્યનું અસ્તિત્વ નથી.

6. જો $bold space bold f bold left parenthesis bold x bold right parenthesis bold space open curly brackets table attributes columnalign left end attributes row cell fraction numerator bold sin bold left parenthesis bold a bold right parenthesis bold 1 bold right parenthesis bold x bold plus bold sinx over denominator bold x end fraction bold comma end cell row cell bold c bold comma end cell row cell fraction numerator square root of bold x bold plus bold bx to the power of bold 2 end root bold space bold minus bold space square root of bold x over denominator bold bx to the power of begin display style bold 3 over bold 2 end style end exponent end fraction bold space bold comma end cell end table close table row cell bold x bold less than bold 0 end cell row cell bold x bold space bold equals bold space bold 0 bold space end cell row cell bold x bold greater than bold 0 end cell end table$ એ x = 0 આગળ સતત હોય તો ........

$bold a bold space bold equals bold space fraction numerator bold minus bold 3 over denominator bold 2 end fraction bold comma bold space bold b bold space bold element of bold space bold R bold comma bold space bold c bold space bold equals bold space bold 1 over bold 2$
$bold a bold space bold equals bold space fraction numerator bold minus bold 3 over denominator bold 2 end fraction bold comma bold space bold b bold space bold equals bold space bold 0 bold comma bold space bold c bold equals bold 1 over bold 2$
$bold a bold space bold equals bold space bold minus bold 3 over bold 2 bold comma bold space bold b bold space bold equals bold space bold 1 bold comma bold space bold c bold space bold equals bold space bold 1 over bold 2$
$bold a bold space bold equals bold space fraction numerator bold minus bold 3 over denominator bold 2 end fraction bold comma bold space bold b bold space bold element of bold space fraction numerator bold minus bold 1 over denominator bold 2 end fraction$

ધારો કે f(x) = g(x) $fraction numerator bold e to the power of begin display style bold 1 over bold x end style end exponent bold space bold minus bold space bold e to the power of bold 1 over bold x end exponent over denominator bold e to the power of bold 1 over bold x end exponent bold space bold plus bold space bold e to the power of bold 1 over bold x end exponent end fraction$ જ્યાં, g સતત વિધેય છે. જો $bold lim with bold x bold rightwards arrow bold 0 below$ f(x) નું અસ્તિત્વ હોય, તો