Help - I have three problems that I am stuck on -
1. Lim x approaches infinity
(x-3/x squared + 4)
2. Lim x approaches 3
x cubed - x squared - 7 x +3/
x squared - 9
3. Lim x approaches negative infinity
(x + square root x squared + 2x)
1. You need to use more parentheses to clarify the mathematical operation. I will assume you mean
1. Lim x -> infinity
(x-3)/(x^2 + 4)
When x approaches infinity, you can neglect the constant terms, and you get
Lim x -> infinity
(x/x^2) = 1/x
As x-> infinity, that approaches zero.
Sure! I can help you with these problems.
1. To solve the limit as x approaches infinity, we need to see what happens to the expression as x becomes very large. In this case, we divide the numerator and denominator by the highest power of x, which is x squared.
So the expression becomes:
Lim x approaches infinity (1 - 3/x)/(1 + 4/x^2)
As x approaches infinity, both the numerator and the denominator tend to 0, so we can let x go to infinity without affecting the value of the expression. Therefore, we can evaluate the limit by substituting in the value of x, which is infinity:
Lim x approaches infinity (1 - 3/infinity)/(1 + 4/infinity^2)
Simplifying further, we get:
Lim x approaches infinity (1 - 0)/(1 + 0)
Therefore, the limit is 1.
2. To solve the limit as x approaches 3, we can use direct substitution. Substituting x = 3 into the expression:
Lim x approaches 3 [(3^3 - 3^2 - 7(3) + 3)/(3^2 - 9)]
Simplifying further, we get:
Lim x approaches 3 [(27 - 9 - 21 + 3)/(9 - 9)]
= Lim x approaches 3 (0/0)
We have an indeterminate form of 0/0, which means we need to further simplify the expression. To do this, we can factorize the numerator and denominator:
Lim x approaches 3 [(x - 1)(x - 3)(x + 3)/(x - 3)(x + 3)]
= Lim x approaches 3 [(x - 1)/(x + 3)]
Now we can substitute x = 3 into the simplified expression:
[(3 - 1)/(3 + 3)]
= 2/6
= 1/3
Therefore, the limit is 1/3.
3. To solve the limit as x approaches negative infinity, we can again use direct substitution. Substituting x = -infinity into the expression:
Lim x approaches -infinity [(x + √(x^2 + 2x))]
As x approaches negative infinity, x^2 and 2x both tend to positive infinity, so the expression inside the square root becomes very large. However, we have to remember that we are subtracting this large value from x, which is a negative infinity.
So, the square root term tends to negative infinity, and subtracting it from x doesn't change the result much. Therefore, the limit can be simplified to:
Lim x approaches -infinity x
Since x approaches negative infinity, the limit is negative infinity as well.
I hope this helps! Let me know if you have any more questions.