Question

Find the answer of limit​

Answer 1

Substitute
`y=\arcsin(x)`, so both
`x` and
`y` are approaching 0.

`\displaystyle \lim_(x\to0) (\sin(p \arcsin(x)))/(x) = \lim_(y\to0) (\sin(py))/(\sin(y))`

Then we have

`\displaystyle \lim_(y\to0) (\sin(py))/(py) \cdot (y)/(\sin(y)) \cdot p = p \cdot \lim_(y\to0) (\sin(py))/(py) \cdot \lim_(y\to0) (y)/(\sin(y)) = \boxed{p}`

where we use the known limit

`\displaystyle \lim_(x\to0)(\sin(ax))/(ax) = 1`

if
`a\\eq0`.

Answer 2

p.

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