A calculus problem by Aly Ahmed

Calculus Level 3

lim ⁡ x → 2 ( x x − 2 ∫ 2 x sin ⁡ t t d t ) \lim_{x\to2} \left( \dfrac x{x-2} \int_2^x \dfrac{\sin t}t \, dt \right) Evaluate the limit above to 3 decimal places.


The answer is 0.909.

Aly Ahmed by Aly Ahmed , 34, Egypt

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1 solution

Let ∫ 2 x sin ⁡ t t d t \displaystyle\int_2^x \dfrac{\sin t}{t}dt be equal to f ( x ) − f ( 2 ) f(x)-f(2) . Then the given limit is 2 f ′ ( 2 ) = 2 × sin ⁡ 2 2 = sin ⁡ 2 2f'(2)=2\times \dfrac{\sin 2}{2}=\sin 2 . When the angle is given in radian measure (it should be), the value is ≈ 0.90929 \approx \boxed {0.90929} .

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