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

The limit of a function $f(x)$ as $x$ approaches a value $a$ is written as $\lim_{x \to a} f(x) = L$ , where $L$ is a real number
For the limit to exist, the left-hand limit $\lim_{x \to a^-} f(x)$ $lim_{x \to a^{-}} f (x)$ and right-hand limit $\lim_{x \to a^+} f(x)$ $lim_{x \to a^{+}} f (x)$ must be equal to $L$ $L$
- If $\lim_{x \to a^-} f(x) \neq \lim_{x \to a^+} f(x)$ , then $\lim_{x \to a} f(x)$ does not exist
One-sided limits are written as $\lim_{x \to a^-} f(x)$ for the left-hand limit and $\lim_{x \to a^+} f(x)$ for the right-hand limit
Limits can be infinite
- If $f(x)$ increases without bound as $x$ approaches $a$ , then $\lim_{x \to a} f(x) = \infty$
- If $f(x)$ decreases without bound as $x$ approaches $a$ , then $\lim_{x \to a} f(x) = -\infty$

Epsilon-Delta Definition of a Limit

The epsilon-delta ( $\varepsilon$ - $\delta$ ) definition of a limit states: $\lim_{x \to a} f(x) = L$ if for every $\varepsilon > 0$ , there exists a $\delta > 0$ such that if $0 < |x - a| < \delta$ , then $|f(x) - L| < \varepsilon$
Graphically, for any $\varepsilon$ -neighborhood $(L-\varepsilon, L+\varepsilon)$ around the limit $L$ , there is always a $\delta$ -neighborhood $(a-\delta, a+\delta)$ around $a$ such that $f(x)$ falls within the $\varepsilon$ -neighborhood whenever $x$ is within the $\delta$ -neighborhood (except possibly at $a$ )
To prove a limit exists using the $\varepsilon$ - $\delta$ definition, assume an arbitrary $\varepsilon > 0$ and demonstrate there exists a $\delta > 0$ that satisfies the definition
The $\varepsilon$ $ε$ - $\delta$ $δ$ definition provides a rigorous proof of a finite limit
- Infinite limits require using the definition with different inequalities

Definition of a Limit, The Limit of a Function · Calculus

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 $\lim_{x \to a} f(x) = f(a)$
Jump discontinuities on the graph indicate the left and right-hand limits are not equal, so the limit does not exist at that point

Definition of a Limit, The Precise Definition of a Limit · Calculus

Numerical Evaluation

Numerically, limits can be approximated using tables of values for $x$ $x$ approaching $a$ $a$
- 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 $\varepsilon$ - $\delta$ definition are needed to verify the limit

Limit vs Function Value

The limit of a function $f(x)$ $f (x)$ as $x$ $x$ approaches $a$ $a$ describes the behavior of the function near $a$ $a$ , but not necessarily at $a$ $a$ 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, $f(x) = (x^2-1)/(x-1)$ is undefined at $x=1$ but $\lim_{x \to 1} f(x) = 2$
If a function is continuous at a point $a$ $a$ , then $\lim_{x \to a} f(x) = f(a)$ $lim_{x \to a} f (x) = f (a)$
- For discontinuous functions, the limit and function value are not equal
Jump discontinuities occur when $\lim_{x \to a^-} f(x) \neq \lim_{x \to a^+} f(x)$ $lim_{x \to a^{-}} f (x) \neq = lim_{x \to a^{+}} f (x)$
- The function may still be defined at $a$ , but the sided limits are not equal so there is no overall limit

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