Let `p_n=a^(P_(n-1))-1,AAn=2,3, ,a n dl e tP_1=a^x-1,` where `a in R^+dot` Then evaluate `("lim")_(xvec0)(P_n)/x`

Evaluate: ("lim")_(xvec0)(1-cos2x)/(x^2)

Evaluate: ("lim")_(xvec0)(1+x)^(os e cx c)

Evaluate: ("lim")_(xvec0)(1-cosm x)/(1-cosn x)

Evaluate: ("lim")_(xvec0)1/xsin^(-1)((2x)/(1+x^2))

Let S_n=1+2+3++n and P_n=(S_2)/(S_2-1)dot(S_3)/(S_3-1)dot(S_4)/(S_4-1)(S_n)/(S_n-1) Where n in N ,(ngeq2)dot Then ("lim")_(xvecoo)P_n=______

Evaluate: ("lim")_(ntooo)(n^p sin^2 (n !))/(n+1)

Let int_x^(x+p)f(t)dt be independent of xa n dI_1=int_0^pf(t)dt ,I_2=int_(10)^(p^n+10)f(z)dz for some p , where n in Ndot Then evaluate (l_2)/(l_1)dot

Given ("lim")_(xvec0)(f(x))/(x^2)=2,w h e r e[dot] denotes the greatest integer function, then (a) ("lim")_(xvec0)[f(x)]=0 (b) ("lim")_(xvec0)[f(x)]=1 (c) ("lim")_(xvec0)[(f(x))/x] does not exist (d) ("lim")_(xvec0)[(f(x))/x] exists

If ("lim")_(xvec0)[1+x+(f(x))/x]^(1/x)=e^3, then find the value of 1n(("lim")_(xvec0)[1+(f(x))/x]^(1/x))i s____

If "^(2n+1)P_(n-1):^(2n-1)P_n=3:5, then find the value of ndot