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goody1
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Hi, can anybody help me with this two limits? I have to prove them by the definition of limit. Thank you in advance.
View attachment 9630 View attachment 9631
View attachment 9630 View attachment 9631
Hi Goody, and welcome to MHB!goody said:Hi, can anybody help me with this two limits? I have to prove them by the definition of limit. Thank you in advance.
Not quite, although you started correctly. The limit in this case is as $x\to\infty$, so you want to see what happens when $x$ gets large. This means that the inequality $\dfrac3{|x+2|}<\varepsilon$ has to hold for all $x$ greater than $N$ (where you think of $N$ as being a large number).goody said:Hi Opalg! Do you think I got it correct?
The limit of a function is the value that a function approaches as the input value gets closer and closer to a specific value. It is denoted by the notation lim f(x) as x approaches a.
To prove a limit by definition, you must show that for any positive number ε (epsilon), there exists a corresponding positive number δ (delta) such that if the distance between the input value x and the limit value a is less than δ, then the distance between the function value f(x) and the limit L is less than ε.
Proving a limit by definition is important because it is the most rigorous way to establish the existence of a limit. It also allows us to understand the behavior of a function near a specific point and make predictions about its behavior.
Some common techniques used to prove a limit by definition include algebraic manipulation, the squeeze theorem, and the epsilon-delta definition of a limit.
Yes, a limit can still be proven by definition even if the function is not continuous. The epsilon-delta definition of a limit does not require the function to be continuous, only that it approaches a specific value as the input value gets closer and closer to a certain value.