Question

If $\underset{x\to 0}{lim}\frac{x^{a}si{n}^{b}x}{sin{x}^c}$ where $a,b,cϵR-\left\{0\right\}$,exists and has non-zero value. Then

• A. $a+c=b$
• B. $a+b=c$
• C. $a=b+c$
• D. $a+b+c=0$

Answer 1

Answer: (B)

$a+b=c$

Since, this limit has to exist and be finite,
we multiply and divide my the corresponding exponents of $x$.
The question transforms to,
$li{m}_{x\to 0}\frac{x^a}{x^a}\times \frac{{sin}^{b}x}{x^b}\times \frac{x^c}{{sin}^{c}x}\times x^{a+b-c}$
For the limit to exist, $a+b-c=0$
Or, $a+b=c$