Sec 1.3_blank.pdf - Section 1.3 The Limit of a Function Brett Geiger Southern Methodist University Math 1337 Calculus I Brett Geiger Section 1.3 The

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Section 1.3: The Limit of a Function Brett Geiger Southern Methodist University Math 1337 - Calculus I Brett Geiger Section 1.3: The Limit of a Function
Limit Intuition Consider the function below: Brett Geiger Section 1.3: The Limit of a Function
Limit Intuition Consider the function below: Technically speaking, the function has a “hole” at x = a , so we know the function is not technically defined at x = a . Brett Geiger Section 1.3: The Limit of a Function
Limit Intuition Consider the function below: Technically speaking, the function has a “hole” at x = a , so we know the function is not technically defined at x = a . Realistically though, this feels a bit silly because it clearly looks like the function value “near x = a ” is about L , whatever that value is. Brett Geiger Section 1.3: The Limit of a Function
Limit Intuition Consider the function below: Technically speaking, the function has a “hole” at x = a , so we know the function is not technically defined at x = a . Realistically though, this feels a bit silly because it clearly looks like the function value “near x = a ” is about L , whatever that value is. Limits help to make this line of thinking rigorous. Brett Geiger Section 1.3: The Limit of a Function
Limit Intuition Consider the function below: Technically speaking, the function has a “hole” at x = a , so we know the function is not technically defined at x = a . Realistically though, this feels a bit silly because it clearly looks like the function value “near x = a ” is about L , whatever that value is. Limits help to make this line of thinking rigorous. The concept of limits is the entire backbone of Calculus, and surprisingly (at least in my opinion), is the most important concept going forward as it is the basis of essentially all definitions. Brett Geiger Section 1.3: The Limit of a Function
Definition of a Limit Definition: Suppose f ( x ) is defined when x is near the number a (but not necessarily at x = a . ). Brett Geiger Section 1.3: The Limit of a Function
Definition of a Limit Definition: Suppose f ( x ) is defined when x is near the number a (but not necessarily at x = a . ). Then, we write lim x a f ( x ) = L and say the limit of f ( x ) , as x approaches a , is L if the function values f ( x ) become arbitrarily close to L as we take x -values arbitrarily close to a on both sides (but not equal to a ). Brett Geiger Section 1.3: The Limit of a Function
Definition of a Limit Definition: Suppose f ( x ) is defined when x is near the number a (but not necessarily at x = a . ). Then, we write lim x a f ( x ) = L and say the limit of f ( x ) , as x approaches a , is L if the function values f ( x ) become arbitrarily close to L as we take x -values arbitrarily close to a on both sides (but not equal to a ). Notes: (a) Notice that the definition does not require f ( x ) to even exist at x = a , only for values near a .

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• Calculus, Limit, Limit of a function, Category theory, One-sided limit