Category: 0. BLOG

If lim⁡x→0sin⁡(ax)sin⁡(2x)=3limx→0sin(2x)sin(ax)=3, then a equals A a = 6 B a = 3 C a = 2 D a =

Evaluate lim⁡x→0sin⁡2xsin⁡5xlimx→0sin5xsin2x A 5/2 B 10 C 0 D 2/5 Explanation For small x, sin⁡kx∼kxsinkx∼kx. So sin⁡2xsin⁡5x∼2x5x=2/5sin5xsin2x∼5x2x=2/5. This uses standard

Evaluate lim⁡x→2×2−4x−2limx→2x−2×2−4 A 0 B 4 C 2 D 8 Explanation Factor x2−4=(x−2)(x+2)x2−4=(x−2)(x+2). Cancel (x−2) for x≠2, so limit becomes

Which statement defines lim⁡x→af(x)=Llimx→af(x)=L A f(x) near L B f(a) equals L C x equals a D f(x) constant Explanation

If LHL = RHL = 5 at x=a, then limit at a equals A 5 B Does not exist C

a Which statement matches “closure under addition”? A Product stays in set B Sum stays in set C Inverse always

For W={(x,y,z)∈R3:x+2y+3z=0}W={(x,y,z)∈R3:x+2y+3z=0}, dim⁡(W)dim(W) is A B. 1 B C. 3 C A. 2 D D. 0 Explanation Explanation: A single

For W={(x,y)∈R2:x+y=0}W={(x,y)∈R2:x+y=0}, which is true? A Not closed addition B Not closed scalar C Subspace of R2R2 D Missing zero

Which subset of R2R2 is a subspace? A Line not through B Line through origin C Circle centered origin D

Which item must exist in a vector space? A B. Prime element B C. Division operation C D. Order relation