Function limits are a key concept in calculus, describing how a function behaves as it approaches a specific point. They help us understand a function's behavior near a value, even if the function isn't defined there.
The definition of a limit involves both left-hand and right-hand limits. If these are equal, the overall limit exists. The epsilon-delta definition provides a rigorous mathematical way to prove limits exist.
Limits of Functions
Definition of a Limit
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The limit of a function f(x) as x approaches a value a is written as limx→af(x)=L, where L is a real number
For the limit to exist, the left-hand limit limx→a−f(x) and right-hand limit limx→a+f(x) must be equal to L
If limx→a−f(x)=limx→a+f(x), then limx→af(x) does not exist
One-sided limits are written as limx→a−f(x) for the left-hand limit and limx→a+f(x) for the right-hand limit
Limits can be infinite
If f(x) increases without bound as x approaches a, then limx→af(x)=∞
If f(x) decreases without bound as x approaches a, then limx→af(x)=−∞
Epsilon-Delta Definition of a Limit
The epsilon-delta (ε-δ) definition of a limit states: limx→af(x)=L if for every ε>0, there exists a δ>0 such that if 0<∣x−a∣<δ, then ∣f(x)−L∣<ε
Graphically, for any ε-neighborhood (L−ε,L+ε) around the limit L, there is always a δ-neighborhood (a−δ,a+δ) around a such that f(x) falls within the ε-neighborhood whenever x is within the δ-neighborhood (except possibly at a)
To prove a limit exists using the ε-δ definition, assume an arbitrary ε>0 and demonstrate there exists a δ>0 that satisfies the definition
The ε-δ definition provides a rigorous proof of a finite limit
Infinite limits require using the definition with different inequalities
Evaluating Limits
Graphical Evaluation
Graphically, the limit L of a function f at x=a can be estimated by observing the y-values that f(x) approaches on the graph as x gets closer to a from both sides
If there is an open circle at (a,f(a)) on the graph, the function is undefined at x=a but the limit still exists if the left and right-hand limits are equal
If there is a closed circle at (a,f(a)), the function is defined at x=a and limx→af(x)=f(a)
Jump discontinuities on the graph indicate the left and right-hand limits are not equal, so the limit does not exist at that point
Numerical Evaluation
Numerically, limits can be approximated using tables of values for x approaching a
If f(x) approaches a single value L from both sides of a, the limit is L
Numerical approximations may not always be conclusive, so other methods like graphing or the ε-δ definition are needed to verify the limit
Limit vs Function Value
The limit of a function f(x) as x approaches a describes the behavior of the function near a, but not necessarily at a itself
It represents what y-value the function gets arbitrarily close to
The value of the function f(a) is the y-value of the function evaluated exactly at x=a, if it is defined
Limits describe what is happening very close to a point, while the function value is exactly at that point
A limit can exist even if the function is not defined at the point
For example, f(x)=(x2−1)/(x−1) is undefined at x=1 but limx→1f(x)=2
If a function is continuous at a point a, then limx→af(x)=f(a)
For discontinuous functions, the limit and function value are not equal
Jump discontinuities occur when limx→a−f(x)=limx→a+f(x)
The function may still be defined at a, but the sided limits are not equal so there is no overall limit