Let the function f(x)={(log ₑ(1+5 x)-log ₑ(1+α x))/x, ; if x ≠ 0; 10, ; if x=0 be continuous at x=0 .
Then α is equal to

Solution

Maths sir · 3 steps

Split the log difference into two standard limits; each one just hands you its coefficient.

1. Continuity condition. For f to be continuous at x = 0, the limit of f(x) as x → 0 must equal f(0) = 10.
2. Standard limit. Use lim(x → 0) ln(1+kx)/x = k. Split the fraction into two such limits, one with k = 5 and one with k = α.

   lim(x → 0)(ln(1+5x)/x - ln(1+α x)/x) = 5 - α

3. So. Set 5 - α = 10, which gives α = -5. The answer is (d).

The idea to keep. Whenever you see ln(1+kx) divided by x near zero, read off k straight away. Then match the limit to the value given at the point.

Answer from the NTA final answer key. Solution by PrepWiser.