azizlwl said:Homework Statement
If f(x)=x2 prove that [tex] \lim_{x \to 2} f(x)= 4[/tex]
Solution given:
We must show that given any ε >0, find δ >0 such that |x2-4|<ε when 0<|x-2|<δ
Choose δ≤1 so that <|x-2|<1
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Confuse between the word 'find' and 'choose'.
To prove a limit, you must show that for any value of ε (epsilon) greater than 0, there exists a corresponding value of δ (delta) such that the distance between the input x and the limit L is less than ε whenever the distance between x and the point of interest a is less than δ.
The definition of a limit is the value that a function approaches as its input approaches a certain point.
Yes, a limit can exist even if the value of the function at the point does not exist. This is known as an "undefined" or "indeterminate" form, and it can occur when the function has a vertical asymptote or a removable discontinuity at the point of interest.
The properties of limits include the limit laws, which state that the limit of the sum, difference, product, or quotient of two functions is equal to the sum, difference, product, or quotient of their respective limits. Other properties include the Squeeze Theorem, which states that if two functions approach the same limit at a point, then any function squeezed between them must also approach that limit.
To use algebra to solve a limit, you can try factoring, simplifying, or manipulating the function algebraically to get it into a form that can be evaluated directly. You can also use techniques such as L'Hôpital's rule or trigonometric identities to simplify the function and find the limit.