IDZ 6.3 – Option 15. Solutions Ryabushko A.P.

Using L'Hopital's rule, it is necessary to find the specified limits (1-5). To calculate these values ​​and estimate the permissible relative error (with an accuracy of two decimal places), you can use the differential (1-2).

Let's calculate the first limit: lim(x → ∞) xln(1+1/x) = lim(x → ∞) ln((1+1/x)^x) = ln(e) = 1

Вычислим второй предел: lim(x → 0) (cosx)^x = exp(lim(x → 0) xln(cosx)) = exp(lim(x → 0) (ln(cosx))/(1/x)) = exp(lim(x → 0) -(sinx/x)/(cosx)) = exp(lim(x → 0) -tanx/x) = 1

Let's calculate the third limit: lim(x → 0) (1-cosx)/(sin^2(x)) = lim(x → 0) (sinx/(sinxcosx))(1-cosx)/(sinx)^2 = lim(x → 0) (1-cosx)/(sinx) * 1/cosx = lim(x → 0) tanx/x * 1/cosx = 1

Let's calculate the fourth limit: lim(x → 0) (1-cos(2x))/(x^2) = lim(x → 0) 2sin^2(x)/(x^2*(1+cos(2x) )) = lim(x → 0) 2sin^2(x)/(x^2*(2cos^2(x))) = 1/2

Let's calculate the fifth limit: lim(x → 0) (sqrt(1+x)-1)/x = lim(x → 0) (sqrt(1+x)-1)(sqrt(1+x)+1)/(x(sqrt(1+x)+1)) = lim(x → 0) x/(x*(sqrt(1+x)+1)) = 1/2

To calculate the sixth expression, you can use the differential: 6.15 √(2.037)2-3/(2.037)2+5 ≈ √(2.037+0.002)^2-3/(2.037+0.002)^2+5 ≈ √(2.037^2 +20,0022.037)-3/2.037^2+5 ≈ √(4.150369)-3/2.037^2+5 ≈ 2.036 with a relative error of about 0.001.

To calculate the seventh expression, you can also use the differential: 7.15 log9.5 ≈ log10-0.05ln(10)/ln(2) ≈ 1-0.05*3.322/0.301 ≈ 0.722 with a relative error of about 0.006.

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IDZ 6.3 – Option 15 represents solutions to problems performed by A.P. Ryabushko. Product description includes two tasks.

The first task (1-5) is to find the specified limits using L'Hopital's rule.

The second task (1-2) consists of approximate calculation of the indicated quantities using a differential and estimation of the allowed relative error with an accuracy of two decimal places. The quantities that need to be calculated are:

6.15 √(2.037)2-3/(2.037)2+5 (square root of (2.037)2-3 divided by (2.037)2+5)

7.15 lg9.5 (decimal logarithm of 9.5)

Solutions to problems are prepared in Microsoft Word 2003 using the formula editor and contain a detailed description of each step of the solution.

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