displaystyle lim x arrow 0 (xa sin b x/ sin (xc)), where a, b, c ∈ R ∼ 0 exists and has nonzero value, then

Question

displaystyle lim x arrow 0 (xa sin b x/ sin (xc)), where a, b, c ∈ R ∼ 0 exists and has nonzero value, then (A) a+c=b (B) b + c = a (C) a+b=c (D) None of these

Tardigrade Admin · Accepted Answer

displaystyle lim x arrow 0 (xa sin b x/ sin xc) = displaystyle lim x arrow 0 xa(( sin x/x))b((xc/ sin xc)) xb-c= displaystyle lim x arrow 0 xa+b-c This limit will have non-zero value if a+b=c.

Q. $x \to 0 lim \frac{x ^{a} sin ^{b} x}{sin ( x ^{c} )}$ , where $a, b, c \in R \sim {0}$ , exists and has nonzero value, then

$a + c = b$

$b + c = a$

$a + b = c$

None of these

$x \to 0 lim \frac{x ^{a} sin ^{b} x}{sin x ^{c}}$
$= x \to 0 lim x^{a} (\frac{sin x}{x})^{b} (\frac{x ^{c}}{sin x ^{c}}) x^{b - c} = x \to 0 lim x^{a + b - c}$
This limit will have non-zero value if $a + b = c$ .

Q. x→0lim​sin(xc)xasinbx​, where a,b,c∈R∼{0}, exists and has nonzero value, then

a+c=b

b+c=a

a+b=c