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CBSE

Gujarati JEE Mathematics : લક્ષ-સાતત્ય અને વિકલન

Multiple Choice Questions

41.

bold જ ો bold space bold f bold left parenthesis bold x bold right parenthesis bold space bold equals open curly brackets table row cell fraction numerator bold 1 bold minus bold sin to the power of bold 3 bold x over denominator bold 3 bold cos to the power of bold 2 bold x end fraction bold comma end cell cell bold x bold less than bold pi over bold 2 end cell row cell bold a bold comma end cell cell bold x bold equals bold pi over bold 2 end cell row cell fraction numerator bold b bold left parenthesis bold 1 bold minus bold space bold sinx bold right parenthesis over denominator bold left parenthesis bold pi bold space bold minus bold space bold 2 bold x bold right parenthesis to the power of bold 2 end fraction end cell cell bold x bold greater than bold pi over bold 2 end cell end table close bold space bold એ bold space bold x bold space bold equals bold space bold pi over bold 2 bold space bold આગળ bold space bold સતત bold space bold હ ો ય bold space bold ત ો bold space bold a over bold b bold equals bold space bold. bold. bold. bold. bold. bold. bold.

42.

bold જ ો bold space bold f bold apostrophe bold left parenthesis bold 2 bold right parenthesis bold space bold equals bold space bold 1 bold space bold ત ો bold space bold lim with bold x bold rightwards arrow bold 0 below bold space fraction numerator bold f bold left parenthesis bold 2 bold space bold plus bold space bold h to the power of bold 2 bold right parenthesis bold space bold minus bold space bold f bold left parenthesis bold 2 bold space bold minus bold space bold h to the power of bold 2 bold right parenthesis over denominator bold 2 bold h to the power of bold 2 end fraction bold space bold equals bold space bold. bold. bold. bold. bold. bold. bold. bold. bold space

1
$bold 1 over bold 2$
2
0

43.

bold જ ો bold space bold f bold left parenthesis bold x bold right parenthesis bold space bold equals bold space fraction numerator bold 2 to the power of bold x bold minus bold 1 over denominator square root of bold 1 bold plus bold ax end root bold minus bold 1 end fraction bold space bold અન ે bold space bold lim with bold x bold rightwards arrow bold 0 below bold space bold f bold left parenthesis bold x bold right parenthesis bold space bold equals bold space bold log subscript bold e bold space bold 4 bold comma bold space bold ત ો bold space bold a bold space bold equals bold space bold. bold. bold. bold. bold. bold. bold space

1
log_e 2-1
log_e 2 + 1
log_e 2

44.

bold lim with bold x bold rightwards arrow bold 0 below bold space bold x to the power of bold 7 bold space open square brackets bold 1 over bold x to the power of bold 3 close square brackets bold space bold equals bold space bold. bold. bold. bold. bold.

0
1
-1
અસ્તિત્વ નથી.

Tips: -

આપણે જાણીએ છીએ કે x - 1 ≤ [x] ≤ x, ∀x ∈ R

$bold 1 over bold x to the power of bold 3 bold space bold minus bold space bold 1 bold space bold less than bold space bold x to the power of bold 7 bold space open square brackets bold 1 over bold x to the power of bold 3 close square brackets bold space bold less-than or slanted equal to bold space bold 1 over bold x to the power of bold 3 bold therefore bold space bold x to the power of bold 7 bold space open parentheses bold 1 over bold x to the power of bold 3 bold space bold minus bold space bold 1 close parentheses bold space bold less-than or slanted equal to bold space bold x to the power of bold 7 bold space open square brackets bold 1 over bold x to the power of bold 3 close square brackets bold space bold less-than or slanted equal to bold space bold x to the power of bold 4 bold comma bold space bold for all bold x bold therefore bold space bold સ ે ન ્ ડવ િ ચ bold space bold પ ્ રમ ે ય bold space bold પરથ ી bold comma bold space bold lim with bold x bold rightwards arrow bold 0 below bold space bold x bold 7 bold space bold space open square brackets fraction numerator begin display style bold 1 end style over denominator begin display style bold x to the power of bold 3 end style end fraction close square brackets bold space bold equals bold space bold 0 bold space$

45.

bold f bold left parenthesis bold x bold right parenthesis bold space bold equals bold space open curly brackets fraction numerator bold log bold left parenthesis bold 1 bold _x bold right parenthesis to the power of bold 1 bold plus bold x end exponent over denominator bold x to the power of bold 2 end fraction bold minus bold 1 over bold x close curly brackets bold space bold ત ો bold space bold lim with bold x bold rightwards arrow bold 0 below bold space bold f bold left parenthesis bold x bold right parenthesis bold space bold equals bold space bold. bold. bold. bold. bold. bold. bold.

0
1
$bold 1 over bold 2$
$bold 1 over bold 4$

46. જો $bold f bold left parenthesis bold x bold right parenthesis bold space bold equals bold space open curly brackets table attributes columnalign left end attributes row cell bold vertical line bold x bold vertical line bold comma end cell row cell bold left square bracket bold x bold right square bracket bold comma end cell end table close table row cell bold x bold less-than or slanted equal to bold 2 end cell row cell bold x bold greater than bold 2 end cell end table$ તો ..............

$bold lim with bold x bold rightwards arrow bold 2 bold plus below$ f(x) = f(2)
$bold lim with bold x bold rightwards arrow bold 2 bold plus below$ f(x) = -2
$bold lim with bold x bold rightwards arrow bold 2 bold minus below$ f(x) = -2
$bold lim with bold x bold rightwards arrow bold 2 below$ f(x) નું અસ્તિત્વ નથી

47.

bold f bold left parenthesis bold x bold right parenthesis bold space bold equals bold space open curly brackets table attributes columnalign left end attributes row cell bold xe to the power of bold minus open parentheses fraction numerator bold 1 over denominator bold vertical line bold x bold vertical line end fraction bold plus bold 1 over bold x close parentheses end exponent end cell row cell bold 0 bold space bold space bold space bold semicolon end cell end table close table row cell bold x bold # bold 0 end cell row cell bold x bold equals bold 0 end cell end table bold space bold ત ો bold space bold f bold space bold એ bold space bold x bold space bold equals bold space bold 0 bold space bold આગળ bold space bold space bold. bold. bold. bold. bold. bold. bold. bold. bold. bold. bold. bold. bold. bold.

વ્યાખ્યાયિત નથી.
વિકલનીય છે.
વિકલનીત નથી પરંતું સતત છે.
વિકલન નથી તથા સતત પણ નથી.

48.

શૂન્યેત્તર વિધેય f એ f(x-y) = $fraction numerator bold f bold left parenthesis bold x bold right parenthesis over denominator bold f bold left parenthesis bold y bold right parenthesis end fraction$ પ્રકારનું છે. જો f'(0) = 5 અને f'(5) = 5, to f(5) = ......