What is the limit of f(x)=1x as x tend to the y-axis limx→0+1x?
What is the limit of f(x)=1/x as x tend to the y-axis limx→0+1/x?
Question 10 options:
0
Not enough information
+∞
−∞
1/X
as x----> 0 from the left, you get −∞
as x----> 0 from the right, you get +∞
see graph:
http://www.wolframalpha.com/input/?i=plot+y+%3D+1%2Fx
To find the limit of the function f(x) = 1/x as x approaches 0 from the positive side, we can use algebraic manipulation along with the concept of limits. The notation limx→0+ (read as 'x approaches 0 from the positive side') indicates that we are only considering values of x that are greater than 0.
First, let's rewrite the function as f(x) = 1/x. Now, as x gets arbitrarily close to 0 from the positive side, we can see that the value of 1/x becomes extremely large. This is because as x approaches 0, the denominator gets closer and closer to zero, causing the fraction to increase without bound.
To formally prove this, we can use the definition of a limit. Let's assume L is the limit of f(x) as x approaches 0 from the positive side. In other words, we have limx→0+1/x = L.
Now, for any positive value M, we need to find a positive number δ such that if 0 < x < δ, the resulting f(x) = 1/x will be greater than M.
To find such a δ, we can start by setting 1/x > M and solve for x:
1/x > M
x < 1/M
So, we can choose δ = 1/M. Now, if 0 < x < 1/M, it follows that 1/x > M, satisfying the conditions for a limit.
Thus, the limit of f(x) = 1/x as x approaches 0 from the positive side is positive infinity (∞).
In summary, the limit of f(x) = 1/x as x approaches 0 from the positive side is ∞.