Evaluate the following limit:
`lim_(nto oo)(sum_(r=1)^(n) sqrt(r)sum_(r=1)^(h)1/(sqrt(r)))/(sum_(r=1)^(n)r)`
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Correct Answer - `8/3`
`lim_(n to oo) (sum_(r=1)^(n)sqrt(r) sum_(r=1)^(n)1/(sqrt(4)))/(sum_(r=1)^(n)r)`
`:.` Limit `=lim_(nto oo) (1/n sum_(r=1)^(n)sqrt(4/n)(1/nsum_(r=1)^(n)sqrt(n/r)))/(1/n sum_(r=1)^(n)r/n)`
`=(int_(0)^(1)sqrt(x)dxint_(0)^(1)(dx)/(sqrt(x)))/(int_(0)^(1)x dx) =8/3`
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Find the following limits. `(i)" "lim_(xto2) (4x)/(x^(3)-3)" "(ii)" "lim_(xto1)(log_(10)x-3)/(3x-2)" "(iii)" "lim_(xto pi) (3+cosx)/(2-sinx)`
In each of the given limits function is defined and contin-uously exists in the NBD of x where we have to find the limit. So, we will find the limits...
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If `lim_(xto0) (f(x))/(sin^(2)x)=8,lim_(xto0) (g(x))/(2cosx-xe^(x)+x^(3)+x-2)=lamda" and "` `lim_(xto0) (1+2f(x))^((1)/(g(x)))=(1)/(e)," then"` The va
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If `lim_(xto0)(f(x))/(sin^(2)x)=8,lim_(xto0)(g(x))/(2cosx-xe^(x)+x^(3)+x-2)=lamda" and "` `lim_(x to 0)(1+2f(x))^((1)/(g(x)))=(1)/(e)," then"` `lim_(x
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If `lim_(xtooo) f(x)` exists and is finite and nonzero and if `lim_(xtooo) {f(x)+(3f(x)-1)/(f^(2)(x))}=3`, then the value of `lim_(xtooo) f(x)" is "`_
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