Respuesta :

Answer:

p.

Step-by-step explanation:

Find limit of the numerator;

limit x --> 0  of  sin(p arcsin x) = 0    as (sin 0= 0)

Denominator -

limit x ---> 0 of x  is also 0

So we have the fraction 0/0  (indeterminate)

So we try applying apply ing L'hopitals Rule - that is differentiate numerator and denominator

We obtain p cos (p arcsin x) / (1 - x^2)/ 1  as the derivative

This gives  us

limit x ---> 0 of p cos (p arcsin x) / (1  - x^2)

= p/1                   (as cos 0 = 1

= p

LammettHash

Substitute [tex]y=\arcsin(x)[/tex], so both [tex]x[/tex] and [tex]y[/tex] are approaching 0.

[tex]\displaystyle \lim_{x\to0} \frac{\sin(p \arcsin(x))}{x} = \lim_{y\to0} \frac{\sin(py)}{\sin(y)}[/tex]

Then we have

[tex]\displaystyle \lim_{y\to0} \frac{\sin(py)}{py} \cdot \frac{y}{\sin(y)} \cdot p = p \cdot \lim_{y\to0} \frac{\sin(py)}{py} \cdot \lim_{y\to0} \frac{y}{\sin(y)} = \boxed{p}[/tex]

where we use the known limit

[tex]\displaystyle \lim_{x\to0}\frac{\sin(ax)}{ax} = 1[/tex]

if [tex]a\neq0[/tex].

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