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

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