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CBSE

Gujarati JEE Mathematics : લક્ષ-સાતત્ય અને વિકલન

Multiple Choice Questions

41. 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

42.

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

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

43.

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

44.

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

45.

bold વ િ ધ ે ય bold space bold f bold left parenthesis bold x bold right parenthesis bold space bold equals bold space bold x bold space fraction numerator bold e to the power of bold left square bracket bold x bold right square bracket bold plus bold vertical line bold x bold vertical line end exponent bold space bold minus bold space bold 2 over denominator bold left square bracket bold x bold right square bracket bold space bold plus bold space bold vertical line bold x bold vertical line bold space end fraction bold space bold હ ો ય 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
-1
અસ્તિત્વ નથી.

46.

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.

47.

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) = ......