For m, n ∈ N, the value of displaystyle lim x arrow 0 ( sin xn/ sin xm) is equal to

Question

For m, n ∈ N, the value of displaystyle lim x arrow 0 ( sin xn/ sin xm) is equal to (A) 1, if n < m (B) 0, if n=m (C) (n/m) (D) 0, if n>m

Tardigrade Admin · Accepted Answer

Given m, n ∈ N Now, displaystyle lim x arrow 0 ( sin xn/ sin xm) = displaystyle lim x arrow 0 (( sin xn/xn) xn/( sin xm)x xm) = displaystyle lim x arrow 0 xn-m=0 text for n>m

Q. For $m, n \in N$ , the value of $x \to 0 lim \frac{sin x ^{n}}{sin x ^{m}}$ is equal to

$1,$ if $n < m$

$0,$ if $n = m$

$\frac{n}{m}$

$0,$ if $n > m$

Given $m, n \in N$
Now, $x \to 0 lim \frac{sin x ^{n}}{sin x ^{m}}$
$= x \to 0 lim \frac{\frac{s i n x ^{n}}{x ^{n}} x ^{n}}{\frac{s i n x ^{m}}{x} x ^{m}}$
$= x \to 0 lim x^{n - m} = 0 for n > m$

Q. For m,n∈N, the value of x→0lim​sinxmsinxn​ is equal to

1, if n<m

0, if n=m

mn​

0, if n>m