lim x->infinity ((sqrt(x^2+5))-(sqrt(x^2+3)))

An alternate proof using sandwich concept:

To find Limitx->infinity {sqrt(x^2+5)-sqrt(x^2+3)}

Let us look at what is sqrt(x^2+5)-sqrt(x^2+3). Let it be called f(x). Let us rationalise the numerator , by writing in numerator and denominator form.

We see that f(x) > 0 for all x , as the square root of greater positive number is greater than that of a smaller positive number.

f(x)={sqrt(x^2+5)-sqrt(x^2+3)} {sqrt(x^2+5)+{sqrt(x^2+3)}/{sqrt(x^2+5)+sqrt(x^2+3)}

={(x^2+5)-(x^2+3)}/{sqrt(x^2+5)+sqrt(x^2+3)}

=2/{sqr(x^2+5)+sqrt(x^2+3)}  <  2/(2x) = 1/x.

Now taking the limits,

Lim x-> infinity f(x) < lim x->infinity (1/x )=0

but f(x)>0 for all x.

Thus 0<f(x)<0. f(x) is sandwiching between zero below and a zero aprroaching value from above as x approaches infininty. Therefore, Lim x->infinity f(x) = 0

Therefore, Limit x->infinity {sqrt(x^2+5)-sqrt(x^2+3)} = 0.

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