If f(x) = (x+1)/(x-1), what is:
i) lim f(x) as x approaches 1
ii) lim f(x) as x approaches ∞
My answer:
i) lim f(x) as x approaches 1 is undefined because 2/0 is undefined
ii) lim f(x) as x approaches ∞ = ∞
(∞+1)/( ∞-1) = ∞/∞ = ∞
I agree with your first part, but not the second
get a "feel" for the numbers,
e.g. let x = 1000
then f(1000) = 1001/999 = appr 1.002
let x = 1,000,000
then f(1000000) = 1000001/999999 = 1.000002..
if you take large negative values you will get appr .9999...
So don't you agree that you get 1 ?
Your answer is partially correct. Let me assist you with the corrections:
i) To find the limit of f(x) as x approaches 1, we can substitute 1 into the function: f(1) = (1 + 1)/(1 - 1) = 2/0. This expression is undefined, which means the limit does not exist.
ii) To find the limit of f(x) as x approaches infinity, we can consider the behavior of the function for large values of x. As x becomes very large, the terms (x+1) and (x-1) become insignificant compared to x. Therefore, we can analyze the leading terms:
lim f(x) as x approaches ∞ = lim (x+1)/(x-1) as x approaches ∞
As x approaches infinity, both (x+1) and (x-1) grow without bound. Thus, we can consider the growth of each term without worrying about the specific values. In this case, both terms grow at the same rate, so we can compare their leading coefficients:
lim f(x) as x approaches ∞ = lim (x+1)/(x-1) as x approaches ∞
By dividing both the numerator and denominator by x, we get:
lim f(x) as x approaches ∞ = lim (1+1/x)/(1-1/x) as x approaches ∞
As x approaches infinity, 1/x approaches 0, giving us:
lim f(x) as x approaches ∞ = lim (1+0)/(1-0) = 1/1 = 1
Therefore, the limit of f(x) as x approaches infinity is 1.
To find the limits of a function, such as f(x) = (x+1)/(x-1), as x approaches a specific value or infinity, we can use algebraic manipulation or substitution.
i) To find lim f(x) as x approaches 1, we substitute x = 1 into the function:
f(1) = (1+1)/(1-1) = 2/0
Since division by zero is undefined, the limit is also undefined.
ii) To find lim f(x) as x approaches infinity, we can observe the behavior of the function as x becomes larger and larger.
As x approaches infinity, the numerator and the denominator both increase without bound. Thus, we can simplify the expression:
lim f(x) as x approaches ∞ = lim (x+1)/(x-1) as x approaches ∞
As x becomes very large, the constant term 1 becomes insignificant when compared to x. So we can assume that x is dominant in the expression:
lim (x+1)/(x-1) as x approaches ∞ ≈ lim (x/x) as x approaches ∞
Canceling out the common factor x gives:
lim f(x) as x approaches ∞ ≈ lim (1+1/x)/(1-1/x) as x approaches ∞
As x approaches infinity, 1/x approaches zero:
lim (1+1/x)/(1-1/x) as x approaches ∞ ≈ (1+0)/(1-0) = 1/1 = 1
Therefore, the limit of f(x) as x approaches infinity is 1.