Limits, Continuity & Differentiability
Limits
nta_pyq_2025_jan
Grade 12
Question:
1 If lim , then \alpha is equal to ________ t \alpha 8 3 t t\to0 (\int (3x + 5) dx) = ( ) 0 5e 5
Step-by-Step Solution
Key Concept: Apply the core result for standard limits and expansions and simplify using the given constraints.
\alpha 1 1 t = exp(lim (\int (3x + 5) dx - 1)) (64) 5e t\to0 t 0 1 t+1 ⎛ 1 (3x + 5) ⎞ = exp lim ( ) - 1 ⎝ t\to0 t 3(t + 1) ⎠ 0 t+1 t+1 1 8 - 5 = exp(lim ( - 1)) t\to0 t 3(t + 1) t+1 t+1 1 8 - 5 - 3t - 3 = exp(lim ( )) t\to0 t 3(t + 1) t+1 t+1 8 ⋅ ln 8 - 5 ln 5 - 3 = exp(lim ( )) t\to0 3(t + 1) 8 5 ln 8 - ln 5 - 3 = exp( ) 5 8 8 2/3 ⎛ ln( ) ⎞ 8 \alpha 5 5 = ( ) = exp⎜ ⎜ - 1⎟ ⎟ 5 5e 5 ⎝ ⎠ 2/3 1/3 1/3 2/3 8 6 2 8 \alpha 8 8 ⋅ 8 64 8 \Rightarrow ( ) = ( ) = ( ) = ( ) 5 3 2 5 5 5 5 ⋅ 5 5 5 \Rightarrow \alpha = 64
Correct Answer: 64