Let f:R−{0}→R be a function such that f(x)−6f(x1)=3x35−25. If the limx→0(αx1+f(x))=β; α,β∈R, then α+2β is equal to
Get the complete, step-by-step math solution for: "Let f: {R} - \{0\} → {R} be a function such that f(x) - 6 f(1)/(x) = (35)/(3x) - (5)/(2). If the _{x → 0} (1)/(α x) + f(x) = β; α, β {R}, then α + 2β ...". Powered by SolveForX AI math tutor.
Step-by-Step Solution
Step 1: Formulate a system of equations for f(x)
We are given an equation involving f(x) and f(1/x). To solve for f(x), we can create a system of two linear equations. The first equation is the given one. The second equation is obtained by replacing x with 1/x in the original equation. This substitution transforms f(x) into f(1/x) and f(1/x) into f(x).
Step 2: Solve for f(x)
To solve for f(x), we multiply equation (2) by 6 and add it to equation (1). This eliminates f(1/x), allowing us to isolate f(x). After combining terms and simplifying, we find the explicit expression for f(x).
Step 3: Evaluate the limit
Now we substitute the derived expression for f(x) into the given limit expression. The limit is given as β. We need to analyze this limit as x approaches 0.
Step 4: Determine α and β
For the limit to exist and be finite (equal to β), the term with 1/x must vanish. This means the coefficient of 1/x must be zero. From this, we can find α. Once α is found, the limit simplifies to the remaining terms as x→0, which gives us β.
Step 5: Calculate α+2β
From the previous step, setting the coefficient of 1/x to zero gives 1/α+1/3=0, which implies α=−3. Then, substituting α=−3 into the limit expression, we get limx→0(6x−1/2)=−1/2, so β=−1/2. Finally, we calculate α+2β.