Find the limit. lim x → 1 + x 2 + x x − 1

Question

Find the limit.       lim          x      →      1            ﻿    +                    x                  2                    +      x              x      −      1

scrapbymarieix · Accepted Answer

The limit is to be taken from the right, that is denoted bythe + sign as a post super script of the value at that the limit should be calculated. That says the limit is above the real numbers.
The nominator is steady and has nulls for x=0 and -1. Since the limit is to be calculated at x=1, the nominator is limited,finite and 2. So close to x=1 there is sequence

x_{n} = 1 + 1 / n^{2}

, such that

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

for sufficient large n. Take m fixed than there are for all n>m for that

n (2 + 3 / n^{2} + 1 / n 4) > m (2 + 3 / m^{2} + 1 / m^{4})

valid. For example m=1: 6 must be exceeded that is valid for all n>=2. For 2 it is 2(2+3/2+1/2)=2*6=12.
The limit does not exist or is

+ \infty

. Proved by an example for that the limit is not finite.

Find the limit. lim_(xrightarrow 1)+(x^2 +x)/(sqrt(x-1))

Answered question

Answer & Explanation

New Questions in Calculus 1