Find Limit of F(x) at x=1: 2x^2, 3, x+

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Since in this case, both the left and right limits are equal to 2, that is the overall limit at x=1. In summary, when looking for the limit of a function at a particular value, we must find a number that the function gets close to as x approaches that value. The value of the function at that point is irrelevant. In this case, the limit of F(x) at x=1 is 2, as the function approaches 2 from both the left and right.
  • #1
bard
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F(x)= 2x^2, x<1
3, X=1
X+, x>1

Find

lim(x-->1) f(X)

f(1)=

not exactly sure how to do this. would not f(1) be just 3, since it is defined as that in the function. I am not sure about how to take that limit.

Thanks
 
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  • #2
When it comes to the limit of a function at x=a, the value of the function at a is irrelevant. In fact the function may not even be defined at "a", but the limit could still exist. So the fact that f(1)=3 is irrelevant to the problem. When looking for the limit, we must find a number that the function gets very close to as x approaches "a". It must get close to the same number when approaching from the left or right, or else the limit does not exist. In the case of this function, I think you might have made an error in typing it when you said that F(x) is equal to "X+" for x>1. But whatever it is, just see what number the function gets close to as x gets close to 1 from the right and the left. If they are the same number, then this is the limit; if not, then the limit does not exist.
 
  • #3
sorry that should be x+1
 
  • #4
ok so

lim(x->1+)=(1+1)=2
lim(x->1-)=(1+1)=2 so the limit exists, i don't understand how that helps in finding the overall limit of F(x) at 1.
 
  • #5
So the limit is 2. The limit is just the number the function approaches from both the left and the right.
 

1. What does it mean to find the limit of a function at a certain point?

Finding the limit of a function at a point means determining the value that the function approaches as the input (x) gets closer and closer to the specified point. This value may or may not be the same as the actual value of the function at that point.

2. How is the limit of a function calculated?

The limit of a function can be calculated algebraically or graphically. Algebraically, you can plug in values that are closer and closer to the specified point and see what value the function approaches. Graphically, you can use the graph of the function to visually see what value the function approaches as x gets closer to the specified point.

3. What is the limit of the function F(x) = 2x^2 at x=1?

The limit of the function F(x) = 2x^2 at x=1 is 2. As x approaches 1, the function approaches a value of 2. You can see this by plugging in values closer and closer to 1, such as 0.9, 0.99, and 0.999, and seeing that the output values get closer and closer to 2.

4. How is the limit of a constant function, such as F(x) = 3, calculated?

The limit of a constant function, such as F(x) = 3, is simply the constant value itself. This is because no matter what value x approaches, the output value will always be 3.

5. Can the limit of a function be different from the actual value of the function at a point?

Yes, the limit of a function can be different from the actual value of the function at a point. This can occur when there is a discontinuity or a hole in the graph of the function at that point. In these cases, the limit represents the value the function approaches from both sides of the discontinuity, while the actual value at the point may not exist or may be different.

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