Write a function that has two infinite limits at two different x-values.
Write a function where both one sided limits exist at some x value but the two sided limit do not.
I don't understand how this would look, please help!
#1 any rational function with vertical asymptotes will do this:
1/(x-1)(x+1)
#2 f(x) =
x+1 for x <= 1
x-1 for x > 1
just any function where there is a finite break at some x value
To understand the concept of functions with different types of limits, let's first define what limits are in the context of calculus.
In calculus, a limit is a fundamental concept that describes the behavior of a function as the input values get arbitrarily close to a certain point. It helps us understand how a function behaves near a specific point or as it approaches infinity or negative infinity.
Now, let's address your first request: a function that has two infinite limits at two different x-values.
Consider the following function:
f(x) = 1 / (x^2)
We can analyze the limits at two different x-values: positive and negative infinity.
1. As x approaches positive infinity (x → +∞):
lim(x → +∞) (1 / (x^2)) = 0
Here, the function approaches zero as x becomes larger and larger, implying that the limit at positive infinity is zero.
2. As x approaches negative infinity (x → -∞):
lim(x → -∞) (1 / (x^2)) = 0
Similarly, as x approaches negative infinity, the function also approaches zero.
Thus, the function f(x) = 1 / (x^2) has two infinite limits at two different x-values.
Moving on to the second request: a function where one-sided limits exist, but the two-sided limit does not.
Consider the function:
g(x) = 1 / x, if x ≠ 0
= 0, if x = 0
1. As x approaches 0 from the positive side (x → 0+):
lim(x → 0+) (1 / x) = +∞
The function approaches positive infinity as x gets arbitrarily close to 0 from the positive side.
2. As x approaches 0 from the negative side (x → 0-):
lim(x → 0-) (1 / x) = -∞
Likewise, the function approaches negative infinity as x gets arbitrarily close to 0 from the negative side.
However, when we consider the two-sided limit (x → 0), it does not exist because the left-sided and right-sided limits approach different values.
In summary, the function g(x) = 1 / x has one-sided limits that exist at x = 0, but the two-sided limit does not exist.
It's important to note that the examples given here are just two of many possible functions demonstrating different types of limits. Mathematically, there are limitless ways to design such functions.