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Maths-

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Question

If $f open parentheses x close parentheses equals vertical line log subscript 10 end subscript invisible function application x vertical line$ , then at $x equals 1$

$f (x)$ is continuous and $f^{}$ 1+=log10⁡e, f'1-=-log10⁡e'/>
$f (x)$ is continuous and $f^{}$ 1+=log10⁡e, f'1-=log10⁡e'/>
$f (x)$ is continuous and $f^{}$ 1-=log10⁡e, f'1+=-log10⁡e'/>
None of these

The correct answer is: $f left parenthesis x right parenthesis$ is continuous and $f to the power of ´ end exponent open parentheses 1 to the power of plus end exponent close parentheses equals log subscript 10 end subscript invisible function application e comma blank f to the power of ´ end exponent open parentheses 1 to the power of minus end exponent close parentheses equals negative log subscript 10 end subscript invisible function application e$

As is evident from the graph of $f left parenthesis x right parenthesis$ that it is continuous but not differentiable at $x equals 1$

Now,
$f to the power of ´ ´ end exponent open parentheses 1 to the power of plus end exponent close parentheses equals stack lim with x rightwards arrow 1 to the power of plus end exponent below invisible function application fraction numerator f open parentheses x close parentheses minus f left parenthesis 1 right parenthesis over denominator x minus 1 end fraction$
$rightwards double arrow f to the power of ´ ´ end exponent open parentheses 1 to the power of plus end exponent close parentheses equals stack lim with h rightwards arrow 0 below invisible function application fraction numerator f open parentheses 1 plus h close parentheses minus f left parenthesis 1 right parenthesis over denominator h end fraction$
$rightwards double arrow f to the power of ´ ´ end exponent open parentheses 1 to the power of plus end exponent close parentheses equals stack lim with h rightwards arrow 0 below invisible function application fraction numerator log subscript 10 end subscript invisible function application open parentheses 1 plus h close parentheses minus 0 over denominator h end fraction$
$rightwards double arrow f to the power of ´ ´ end exponent open parentheses 1 to the power of plus end exponent close parentheses equals stack lim with h rightwards arrow 0 below invisible function application fraction numerator log invisible function application left parenthesis 1 plus h right parenthesis over denominator h. log subscript e end subscript invisible function application 10 end fraction equals fraction numerator 1 over denominator log subscript e end subscript invisible function application 10 end fraction equals log subscript 10 end subscript invisible function application e$
$f to the power of ´ ´ end exponent open parentheses 1 to the power of minus end exponent close parentheses equals stack lim with x rightwards arrow 1 to the power of plus end exponent below invisible function application fraction numerator f open parentheses x close parentheses minus f left parenthesis 1 right parenthesis over denominator x minus 1 end fraction$
$rightwards double arrow f to the power of ´ ´ end exponent open parentheses 1 to the power of minus end exponent close parentheses equals stack lim with h rightwards arrow 0 below invisible function application fraction numerator f open parentheses 1 minus h close parentheses minus f left parenthesis 1 right parenthesis over denominator h end fraction$
$rightwards double arrow f to the power of ´ ´ end exponent open parentheses 1 to the power of minus end exponent close parentheses equals stack lim with h rightwards arrow 0 below invisible function application fraction numerator log subscript 10 end subscript invisible function application open parentheses 1 minus h close parentheses over denominator h end fraction equals stack lim with h rightwards arrow 0 below invisible function application fraction numerator log subscript e end subscript invisible function application left parenthesis 1 minus h right parenthesis over denominator h log subscript e end subscript invisible function application 10 end fraction equals negative log subscript 10 end subscript invisible function application e$

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