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ما الفرق بين النهايات والدوال المتصلة؟ 😕
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✅ Answered💬 5 replies👁 218 views
ما الفرق بين النهايات والدوال المتصلة؟ انا مش فاهم حاجه 😕
@amine_67
Okay, let's break this down. A limit is the value that a function approaches as the input approaches a certain point. For example, if we have the function f(x) = x^2, the limit as x approaches 2 is 4. But continuity is different. A function is continuous at a point if three conditions are met: 1) the function is defined at that point, 2) the limit exists as x approaches that point, and 3) the limit equals the function's value at that point. So, for f(x) = x^2, it's continuous at x=2 because f(2)=4 and the limit as x approaches 2 is also 4. But if we have a function like f(x) = (x^2 - 4)/(x - 2), it's not defined at x=2, so it's not continuous there, but the limit as x approaches 2 is 4. Hope that helps!
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@sarahh.music
just google it lol
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@luca.s
limits are like when you're trying to reach the fridge but your mom keeps stopping you 😂
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@fatima_03
wait but what about functions that have limits but aren't continuous? like what's the difference then?
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@just_moi_06
i had this on my exam last year, just remember that continuity needs the function to be defined at the point and the limit to equal the function value there. limits can exist even if the function isn't defined there.
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@carlos99
hey @luca.s, your example is funny but not really accurate. limits are about the behavior of the function as you approach a point, not necessarily reaching it. 😅
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@yuki.study
but how do you know if a function is continuous or not? like what steps do you take?
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@omar_dz
let's take an example. consider f(x) = (x^2 - 1)/(x - 1). the limit as x approaches 1 is 2, but f(1) is undefined. so the limit exists but the function isn't continuous at x=1. but if we have f(x) = x^2, it's continuous everywhere because the limit equals the function value at every point.
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Original Poster
omg thanks @amine_67 and @omar_dz!! that makes so much more sense now 😊
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Original Poster
wait but what about functions that have limits but aren't continuous? like can you give another example?
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@fatima_03
sure! think about f(x) = |x|. the limit as x approaches 0 is 0, and f(0) = 0, so it's continuous at x=0. but if you have f(x) = 1/x, the limit as x approaches 0 doesn't exist (it goes to infinity), so it's not continuous there.
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@just_moi_06
another example is f(x) = (sin x)/x. the limit as x approaches 0 is 1, but f(0) is undefined. so the limit exists but the function isn't continuous at x=0.
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