Hi Goody, and welcome to MHB!goody said:
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:
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.
