Limits
Assertion-reason limit comparison
nta_pyq_2025_apr
Grade 12
Question:
Given below are two statements : -1$1+x$tan$x+log$$\sqrt$-2x e Statement I : lim$1-x$2 x$\to$0 ($) = x$5 5 2 Statement II : lim x$\to$1 (x$1-x$$) = e$1 2 in the light of the above statements, choose the correct answer from the options given below :
Statement I is false but Statement II is true
Statement I is true but Statement II is false
Both Statement I and Statement II are false
$Both Statement I and Statement II are true 1$
Step-by-Step Solution
Key Concept: Expand the expression near the limiting point using the appropriate standard trigonometric, logarithmic, or exponential approximation.
$tan -1$$x + 1$2 [$l_n$($1 + x)$-$l_n$($1 - x)$] - 2x lim (4) x$\to$0 x 5 3 5 2 3 2 3 x x 1 x x x x (x - +$\ldots$) + [x - +$\ldots$-$(-x$- -$\ldots$)] - 2x 3 5 2 2 3 2$3 = lim$5 x$\to$0 x 5 2x 2x +$\ldots$- 2x 5$2 = lim$= x$\to$0 x 5 5 2 2 limx$\tol$($)(x-1)$$(1-x)$$(1-x) -2$lim$x = e = e$x$\to$1 $\Rightarrow$ Both statements correct 1
Correct Answer: 4