If limx→1(x−1)3(x−1)(6+λcos(x−1))+μsin(1−x)=−1 where λ,μ∈R then λ+μ is equal to:
Get the complete, step-by-step math solution for: "If _{x 1} ((x-1)(6 + cos(x-1)) + sin(1-x))/((x-1)³) = -1 where {R} then + is equal to:". Powered by SolveForX AI math tutor.
Step-by-Step Solution
Step 1: Substitute t=x−1
To simplify the limit expression, we introduce a substitution. Let t=x−1. As x approaches 1, t approaches 0. This transformation makes the limit easier to evaluate using standard Maclaurin series expansions.
Step 2: Rewrite the limit in terms of t
After substituting t=x−1, the expression becomes t3t(6+λcos(t))+μsin(−t). We also use the property that sin(−t)=−sin(t).
Step 3: Apply Maclaurin series expansions
We use the Maclaurin series expansions for cos(t)=1−2t2!+4t4!−… and sin(t)=t−3t3!+5t5!−…. We only need terms up to t3 in the numerator to match the denominator.
Step 4: Simplify the expression
Expand the terms in the numerator and group them by powers of t. This helps us identify the coefficients of t and t3.
Step 5: Equate coefficients to solve for λ and μ
For the limit to be finite and non-zero, the coefficient of t in the numerator must be zero. This gives us 6+λ−μ=0. Then, the limit becomes the coefficient of t3, so 6μ−2λ=−1.
Step 6: Solve the system of equations
Substitute the first equation into the second: (λ+6)−3λ=−6. This simplifies to −2λ=−12, so λ=6. Then, μ=6+6=12.
Step 7: Calculate λ+μ
Finally, we calculate the sum of λ and μ using the values we found.