what are the limits?

Question

what are the limits of....

a) lim ((SQRT(x^2 +x)) -x)
x->inf

b) lim (x*sinx)/(x^2 +5)
x->inf

"SQRT" is "the square root of"...

Answer 1

a) For every x, SQRT(x^2 +x)) -x = x SQRT(1 +1/x) - x = x[SQRT(1 +1/x) - 1] = x[(1 +1/x)^(1/2) - 1]. We know that, if u --> 0, then (1 + u)^m = 1 + u/m + o(u), where o is a function such that o(u)/u --> o as u --> 0. Since 1/x --> 0 as x --> oo, x[(1 +1/x)^(1/2) - 1] = x [1 +(1/2x) + o(1/x) -1] = 1/2 + o(1/x)/(1/x), so that this expression goes to 1/2 when x --> oo. The limit is, therefore, 1/2.

b) For every x, (x*sinx)/(x^2 +5) = sin(x)/(x + 5/x). The sine function is bounded, because it's values are in [-1, 1]. And x + 5/x goes to oo as x goes. So, lim (x --> oo) sin(x)/(x + 5/x) = 0. The limit is 0.

Answer 2

reckoning on how constrained the vocabulary, each little thing you do and each interplay with others would be affected. somebody with a severely constrained vocabulary might have a complicated time getting a stable interest, expertise others, cashing in on coaching. there is not any section in existence which will get away being touched by using this actual lack. in case you do no longer understand a thank you to precise your self, how are you able to're making somebody understand? in case you won't have the capacity to understand what somebody is telling you, how are you able to have interaction in a verbal substitute? in case you do no longer carry close the which skill of the words, how are you going to be waiting to stick to instructions? no longer basically will this reason very a great number of frustration, however the guy's vanity will go through an endless deal via fact of it. (people who won't be able to examine go through the main. while somebody with a small vocabulary can properly suited this by using analyzing and searching up words and getting to understand, the only which hasn't been taught to examine is at a particular disadvantage.) Will the skill to think of (or creativity) be constrained via a small vocabulary? I don’t think of so, however the expression of that inner image would be; the skill to talk it to others is lessened. (this is frequently complicated adequate to of course present day recommendations to others without the extra barrier imposed by using a constrained vocabulary.) That, in turn, finally ends up in extra frustration. i assume i'm no longer saying something new, as a techniques via fact the different solutions circulate, yet those are a number of my ideas on your question.

Answer 3

If X approaches infinity there should be no limit.

Answer 4

yikes!