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Which of the following function is not continuous at x = 0?A. `f(x)=(1+2x)^(1//x),xne0` `=e^(2),x=0`B. `f(x)=sinx-cosx,xne0` =-1,x=0C. `f(x)=(e^(1//x)-1)/(e^(1//x)),xne0` `=-1,x=0`D. `f(x)=(e^(5x)-e^(2x))/(sin3x),xne0` `=1,x=0` |
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Answer» Correct Answer - C Option (a), Here ,f(0)=`e^(2)` and `underset( x to 0)(lim) f(x) = underset(x to 0)(lim) (1+2x)^(1//x) = e^(2)` `[because underset( x to 0)(lim)(1 + lambda x)^(1//x)= e^(lambda)]` `therefore " "f(0) = underset(x to0)(lim) f(x) = e^(2)` ` rArr ` f (x) is continous at x = 0 option (b) Here, f(0) = -1 and `underset(x to 0) (lim) f(x) = underset(x t0)(lim) (sin x - cos x) = sin 0 - cos 0 = - 1` `therefore " "f(0) = underset(x to0)(lim) f(x) =- 1` `rArr` f(x) is continuous at x = 0 option (c), Here, `f(0)=- 1` and `underset( x to 0)(lim)f(x) = underset(x to 0 )(lim) (e^(1//x) -1)/(e^(1//x) +1) = underset(x to 0)(lim)(1-e^(1//x))/(1+e^(-1//x)) = 1` `therefore f(0) ne underset(xto0)(lim) f(x) rArr f(x)` f(x) is is not continous option (d) . Here f(0) =1 and `underset(x to 0)(lim) f(x) =underset(x to 0)(lim) (e^(5x) -e^(2x))/(sin 3x) xx (3x)/(3x)` ` = underset(x to0)(lim)(e^(5x )-e^(2x))/( 3x) xx underset(x to 0)(lim) (3x)/(sin3x)` `(1+(5x)/(1!) +((5x)^(2))/(2!) + .....)- = underset(x to 0)(lim)((1+(2x)/(1!)+((2x)^(2))/(2!)........))/(3x)` `(because underset(x to)(lim) (3x)/(sin x3) = 1)` `= underset(x to 0)(lim) (((3)/(1!) +(21x^(2))/(2!)+....))/(3x) = 1` `therefore " "f(0) = underset(x to 0)(lim) f(x) = 1` `rArr f(x)` is continouous at x = 0 |
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