This page helps you learn how to find the limit of a function in an analytic way.

[x + [x + x^{1/2}]^{1/2}]^{1/2}/[x + 1]^{1/2}

when x tends to +∞

lim [x + [x + x^{1/2}]^{1/2}]^{1/2}/[x + 1]^{1/2} =
lim [1 + (1/x + x^{-3/2})]^{1/2}/[1 + 1/x]^{1/2} = 1

[x^{1/2} + x^{1/3} + x^{1/4}]/[2x + 1]^{1/2}

when x tends to +∞

lim [x^{1/2} + x^{1/3} + x^{1/4}]/[2x + 1]^{1/2} =

lim x^{1/2}[1 + x^{-1/6} + x^{-1/4}]/[x^{1/2}(2 + 1/x)^{1/2}] =

lim [1 + x^{-1/6} + x^{-1/4}]/(2 + 1/x)^{1/2} = 1/2^{1/2}

lim x

lim [1 + x

[(1 + 2x)^{1/2} - 3]/[x^{1/2} - 2]

when x tends to 4

lim [(1 + 2x)^{1/2} - 3]/(x^{1/2} - 2) = d'H = (1 + 2x)^{-1/2}/(1/2 x^{-1/2}) =

2x^{1/2}/(2x + 1)^{1/2} = 4/3

2x

[(1 - x)^{1/2} - 3]/[2 + x^{1/3}]

when x tends to -8

lim [(1 - x)^{1/2} - 3]/(2 + x^{1/3}) = d'H =

[-1/2 (1 - x)^{-1/2}]/(1/3 x^{-2/3}) = -2

[-1/2 (1 - x)

[x^{1/2} - a^{1/2} + (x - a)^{1/2}]/(x^{2} - a^{2})^{1/2}

when x tends to a, a > 0

lim [x^{1/2} - a^{1/2} + (x - a)^{1/2}]/(x^{2} - a^{2})^{1/2} =

lim [x^{1/2} - a^{1/2} + (x - a)^{1/2}]/[(x - a)(x + a)]^{1/2} =

lim [x^{1/2} - a^{1/2}]/[(x - a)(x + a)]^{1/2} + 1/[x + a]^{1/2}=

lim [x - a]/[(x - a)^{1/2}(x + a)^{1/2}(x^{1/2} + a^{1/2})] + 1/[x + a]^{1/2}=

lim (x - a)^{1/2}/[(x + a)^{1/2}(x^{1/2} + a^{1/2})] + 1/[x + a]^{1/2} = 1/[2a]^{1/2}

lim [x

lim [x

lim [x - a]/[(x - a)

lim (x - a)

If you want to find more analytical examples how to calculate limits go to the page function - limits - analytical examples.

To train another analytical examples containing limits calculations go to the page series - convergence - analytical examples.