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Question
L
i
m
x
→
0
1
-
cos
x
x
2
এর মান কত?
0
1
1
2
-1
ANSWER : 3
Descrption
<p>L'Hospital's Rule ব্যবহার করে সমাধান করা হলো-</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow></munder><mfrac><mrow><mn>1</mn><mo>-</mo><mi>cos</mi><mi>x</mi></mrow><msup><mi>x</mi><mn>2</mn></msup></mfrac><mspace linebreak="newline"> </mspace><mspace linebreak="newline"> </mspace><mo>=</mo><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow></munder><mfrac><mrow><mi>sin</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mrow><mn>2</mn><mi>x</mi></mrow></mfrac><mspace linebreak="newline"> </mspace><mspace linebreak="newline"> </mspace><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow></munder><mfrac><mrow><mi>sin</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mi>x</mi></mfrac><mspace linebreak="newline"> </mspace><mspace linebreak="newline"> </mspace></math></p><p> </p><p>যেহেতু, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>≤</mo><mfrac><mrow><mi>sin</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mi>x</mi></mfrac><mo>≤</mo><mn>1</mn></math> এবং</p><p><img 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alt="limit as x rightwards arrow 0 of cos left parenthesis x right parenthesis equals limit as x rightwards arrow 0 of 1 equals 1"></p><p> </p><p>সুতরাং squeeze theorem অনুযায়ী, <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>2</mn></mfrac><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow></munder><mfrac><mrow><mi>sin</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mi>x</mi></mfrac><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>×</mo><mn>1</mn><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></math></p><p> </p><p> </p><p> </p>
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