If `lim_(xto1)(asin(x-1)+bcos(x-1)+4)/(x^(2)-1)=-2`, then `|a+b|` is__________.

Question

If `lim_(xto1)(asin(x-1)+bcos(x-1)+4)/(x^(2)-1)=-2`, then `|a+b|` is__________.

If `lim_(xto1)(asin(x-1)+bcos(x-1)+4)/(x^(2)-1)=-2`, then `|a+b|` is__________.
A. does not exist (in R)
B. is equal to 0
C. is equal to 15
D. is equal to 120

Monjur Alahi

4 views

2025-01-04 04:42

1 Answers

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Salim Ahmed Kawsar · Accepted Answer · 2023-02-05T15:29:48+06:00

Correct Answer - `(8)`
Since RHS is finite quantity, at `xto1`, numerator must be 0. Therefore,
`0+b+4=0`
`b=-4`
Then `underset(xto1)lim(asin(x-1)-4cos(x-1)+4)/((x^(2)-1))=-2`
Put `x=1+h`. Then `underset(hto0)lim(asin h+4(1-cos h))/(h(2+h))=-2`
or `underset(hto0)lim(a((sin h)/(h))+4((1-cos h)/(h)))/(2+h)=-2`
or `" "(a(1)+0)/(2)=-2`
or `a=-4`
or `|a+b|=8`