Find lim x approches -1 (sqrt(pi)-sqrt(acos(x)))/sqrt(x+1).

Question

Find lim x → -1 (√π - √cos^-1x)⁄√x+1

Answer 1

since arccos(-1) = pi, the attempt to evaluate this limit by simple substitution gives numerator 0 and also denominator 0; therefore l'Hopital's rule can be applied.

 

Thus we will be taking the limit, as x approaches -1, of an expression having numerator

(-½)(-1/sqrt(1-x²)(arccos x)^-½  and denominator ½(x+1)^-½.

 

Simplified, this means take the limit (x→ -1)  of   (x+1)^½(arccos x)^-½ / sqrt(1-x²).  ⇐REVISED (one factor was lost)

 

Factoring the denominator gives  (arccos x)^-½ (1-x)^½ • (1+x)^½ and now cancellation occurs so that we just take 

the limit (x→ -1)  of (arccos x)^-½/(1-x)^½ which is 1/square root of (two pi).

 

ABOVE ANSWER is revised