Explain why a limit does not exist when 𝑥 approaches to 2 for 𝑓(𝑥) = √2 − 𝑥
assuming you mean √(2-x),
because f(x) is undefined for x > 2
I would say the limit does not exist when x approaches to 2 for that equation is because thing of the x approaching 2.
It can either come from left or right (Positive side or Negative side)
so plugging that in the equation it will look something like this.
**Understand that the negative sign is not an actual sign but just letting you know its from the left side approaching 2
Positive side: f(+2)=√2-(2+) = √-0
Negative side: f(-2)=√2-(2-) = √+0
Now √-0 and √+0 are very different. The √-0 is not possible because of the -(sign) you are not able to square root a negative number, while √+0 is possible so it would = 0.
To determine whether the limit exists for 𝑓(𝑥) = √2 − 𝑥 as 𝑥 approaches 2, we evaluate the left and right limits separately.
First, let's consider the left limit as 𝑥 approaches 2 from the left side (i.e., 𝑥 < 2). We substitute values of 𝑥 that are slightly less than 2 into the function:
lim (𝑥→2-) √2 − 𝑥 = √2 − 2
Since √2 - 2 is a constant value, the left limit is equal to this constant: √2 − 2.
Next, let's consider the right limit as 𝑥 approaches 2 from the right side (i.e., 𝑥 > 2). Again, we substitute values of 𝑥 that are slightly greater than 2 into the function:
lim (𝑥→2+) √2 − 𝑥 = √2 − 2
Once again, we obtain the same constant value of √2 − 2.
Since the left limit and the right limit both equal to √2 − 2, we can conclude that the limit exists if and only if both the left and right limits are equal.
However, in this case, the left and right limits are not equal (√2 − 2 ≠ √2 − 2). Therefore, the limit does not exist as 𝑥 approaches 2 for the function 𝑓(𝑥) = √2 − 𝑥.
To determine if a limit exists as x approaches a particular value, we need to consider the behavior of the function on both sides of that value. In this case, we are looking at the limit as x approaches 2 for the function f(x) = √2 - x.
To find the limit, we need to evaluate the function as x approaches 2 from both the left and the right side of 2 and see if the values converge towards a single value.
Let's determine the limit as x approaches 2 from the left side. We can select values of x that are less than 2, such as 1.9, 1.99, and 1.999, and substitute them into the function to see what happens:
f(1.9) = √2 - 1.9 ≈ -0.071
f(1.99) = √2 - 1.99 ≈ -0.014
f(1.999) = √2 - 1.999 ≈ -0.002
As we approach 2 from the left side, the function values are converging towards √2 - 2. However, we need to consider the behavior from the right side as well.
Let's determine the limit as x approaches 2 from the right side. We can select values of x that are greater than 2, such as 2.1, 2.01, and 2.001, and substitute them into the function:
f(2.1) = √2 - 2.1 ≈ -0.105
f(2.01) = √2 - 2.01 ≈ -0.014
f(2.001) = √2 - 2.001 ≈ -0.001
As we approach 2 from the right side, the function values are also converging towards √2 - 2.
However, notice that the function values from both sides are converging towards different values. From the left side, the values are approaching √2 - 2, while from the right side, they are approaching √2 - 2 as well. Since the function values do not converge to the same value regardless of the direction of approach, we can conclude that the limit does not exist as x approaches 2 for the function f(x) = √2 - x.