Question
limx->infinity 1/(x^2+4x+5)
Answers
GPT 3.5
To find the limit as x approaches infinity of 1/(x^2+4x+5), we can use the fact that as the value of x increases without bound, the higher order terms become negligible compared to the highest power of x.
The highest power of x in the denominator is x^2. Therefore, we divide the expression by x^2 to get:
1/(x^2+4x+5) / x^2 = 1/(x^2(x^2/x^2+4x/x^2+5/x^2))
= 1/(1+4/x+5/x^2)
Taking the limit as x approaches infinity, the terms with 4/x and 5/x^2 become infinitely small compared to 1, so they approach zero. Therefore, we are left with the limit of 1/(1+0+0) as x approaches infinity, which is equal to 1.
Therefore, the limit of 1/(x^2+4x+5) as x approaches infinity is 1.
The highest power of x in the denominator is x^2. Therefore, we divide the expression by x^2 to get:
1/(x^2+4x+5) / x^2 = 1/(x^2(x^2/x^2+4x/x^2+5/x^2))
= 1/(1+4/x+5/x^2)
Taking the limit as x approaches infinity, the terms with 4/x and 5/x^2 become infinitely small compared to 1, so they approach zero. Therefore, we are left with the limit of 1/(1+0+0) as x approaches infinity, which is equal to 1.
Therefore, the limit of 1/(x^2+4x+5) as x approaches infinity is 1.