Question

Lim (x approaches1) [(ln(1+x) - ln2)(3.4x-1 - 3x)] / [(7+x)1/3 - (1 + 3x)1/2]sin(x-1)

Answer 1

SOLUTION

TO EVALUATE

$\displaystyle \sf{\lim_{x \to 1} \: \frac{ (\ln \: (1 + x) - \ln 2)(3. {4}^{x - 1} - 3x)}{ \bigg[{(7 + x)}^{ \frac{1}{3} } - {(1 + 3x)}^{ \frac{1}{2} } \bigg]. \sin (x - 1) }}$

EVALUATION

Here the given expression is

$\displaystyle \sf{\lim_{x \to 1} \: \frac{ (\ln \: (1 + x) - \ln 2)(3. {4}^{x - 1} - 3x)}{ \bigg[{(7 + x)}^{ \frac{1}{3} } - {(1 + 3x)}^{ \frac{1}{2} } \bigg]. \sin (x - 1) }}$

$\displaystyle \sf{ = \lim_{x \to 1} \: \frac{3. \ln \bigg( \frac{1 + x}{2} \bigg)({4}^{x - 1} - x)}{ \bigg[{(7 + x)}^{ \frac{1}{3} } - {(1 + 3x)}^{ \frac{1}{2} } \bigg]. \sin (x - 1) }}$

$\displaystyle \sf{ = \lim_{x \to 1} \: \frac{3. \ln \bigg( 1 + \frac{ x - 1}{2} \bigg)({4}^{x - 1} - x)}{ \bigg[{(7 + x)}^{ \frac{1}{3} } - {(1 + 3x)}^{ \frac{1}{2} } \bigg]. \sin (x - 1) }}$

$\displaystyle \sf{ = \frac{3}{2} . \lim_{x \to 1} \: \: \frac{\ln \bigg( 1 + \frac{ x - 1}{2} \bigg)}{\frac{ x - 1}{2}} \: \lim_{x \to 1} \: \frac{x - 1}{\sin (x - 1)} \:. \: \lim_{x \to 1} \: \frac{ ({4}^{x - 1} - x)}{ \bigg[{(7 + x)}^{ \frac{1}{3} } - {(1 + 3x)}^{ \frac{1}{2} } \bigg] }}$

$\displaystyle \sf{ = \frac{3}{2} . \lim_{x \to 1} \: \: \frac{\ln \bigg( 1 + \frac{ x - 1}{2} \bigg)}{\frac{ x - 1}{2}} \: \lim_{x \to 1} \: \frac{x - 1}{\sin (x - 1)} \:. \: \lim_{x \to 1} \: \frac{({4}^{x - 1} - x)}{ \bigg[{(7 + x)}^{ \frac{1}{3} } - {(1 + 3x)}^{ \frac{1}{2} } \bigg] }}$

$\displaystyle \sf{ = \frac{3}{2} . 1.1 \:. \: \lim_{x \to 1} \: \frac{({4}^{x - 1} - x)}{ \bigg[{(7 + x)}^{ \frac{1}{3} } - {(1 + 3x)}^{ \frac{1}{2} } \bigg] }}$

$\displaystyle \sf{ = \frac{3}{2} \lim_{x \to 1} \: \frac{({4}^{x - 1} \ln 4 - 1)}{ \bigg[ \frac{ 1}{3}. {(7 + x)}^{ \frac{ - 2}{3} } - \frac{3}{2} {(1 + 3x)}^{ \frac{ - 1}{2} } \bigg] }}$

$\displaystyle \sf{ = \frac{3}{2} \: \frac{({4}^{0} \ln 4 - 1)}{ \bigg[ \frac{ 1}{3}. {(7 + 1)}^{ \frac{ - 2}{3} } - \frac{3}{2} {(1 + 3)}^{ \frac{ - 1}{2} } \bigg] }}$

$\displaystyle \sf{ = \frac{3}{2} \: \frac{({4}^{0} \ln 4 - 1)}{ \bigg[ \frac{ 1}{3}. {8}^{ \frac{ - 2}{3} } - \frac{3}{2}. {4}^{ \frac{ - 1}{2} } \bigg] }}$

$\displaystyle \sf{ = \frac{3}{2} \: \frac{(\ln 4 - 1)}{ \bigg[ \frac{ 1}{3}. \frac{1}{4} - \frac{3}{2}. \frac{1}{2} \bigg] }}$

$\displaystyle \sf{ = - \frac{3 6}{16} . \ln \bigg( \frac{4}{e} \bigg)\: }$

$\displaystyle \sf{ = - \frac{9}{4} . \ln \bigg( \frac{4}{e} \bigg)\: }$

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Answer 2

Answer:

Step-by-step explanation: