When evaluating limits graphically, we examine the behavior of a function as we approach a specific x-value from both directions. The key is to observe the y-values as we get arbitrarily close to our target x-value, but not necessarily at that exact point. A function can have a limit at a point even if it's not defined there.
For example, consider a function with a hole at \(x = 2\). As we approach \(x = 2\) from either direction, if the y-values consistently approach 5, then \[\lim_{x \to 2} f(x) = 5\] regardless of whether \(f(2)\) is defined or equals 5.
The fundamental properties of limits provide a structured approach to evaluating complex limits:
Sum Rule: \[\lim_{x \to a}[f(x) + g(x)] = \lim_{x \to a}f(x) + \lim_{x \to a}g(x)\]
Product Rule: \[\lim_{x \to a}[f(x)g(x)] = \lim_{x \to a}f(x) \cdot \lim_{x \to a}g(x)\]
Quotient Rule: \[\lim_{x \to a}\frac{f(x)}{g(x)} = \frac{\lim_{x \to a}f(x)}{\lim_{x \to a}g(x)}\] when \(\lim_{x \to a}g(x) \neq 0\)
Constant Multiple Rule: \[\lim_{x \to a}[c \cdot f(x)] = c \cdot \lim_{x \to a}f(x)\]
Direct substitution works for continuous functions. For example: \[\lim_{x \to 2}(x^2 + 3) = 2^2 + 3 = 7\]
For indeterminate forms like \(\frac{0}{0}\), we use algebraic techniques. For instance:
\[\lim_{x \to 3}\frac{x^2 - 9}{x - 3} = \lim_{x \to 3}\frac{(x+3)(x-3)}{x-3} = \lim_{x \to 3}(x+3) = 6\]
For trigonometric limits, we often use the fundamental limit:
\[\lim_{x \to 0}\frac{\sin(x)}{x} = 1\]
If \(g(x) \leq f(x) \leq h(x)\) for all x near a (except possibly at a), and
\[\lim_{x \to a}g(x) = \lim_{x \to a}h(x) = L\]
then \[\lim_{x \to a}f(x) = L\]
Example: Proving \[\lim_{x \to 0}x\sin(\frac{1}{x}) = 0\]
Since \(-|x| \leq x\sin(\frac{1}{x}) \leq |x|\) and \[\lim_{x \to 0}|x| = 0\]
For rational functions as \(x \to \infty\), divide by highest power:
\[\lim_{x \to \infty}\frac{3x^2 + 2x}{x^2 - 1} = \lim_{x \to \infty}\frac{3 + \frac{2}{x}}{1 - \frac{1}{x^2}} = 3\]
Vertical asymptotes occur where denominator is zero. For example:
\[\lim_{x \to 2^-}\frac{1}{x-2} = -\infty \quad \text{and} \quad \lim_{x \to 2^+}\frac{1}{x-2} = \infty\]
A function is continuous at a point \(a\) if:
\(f(a)\) is defined
\(\lim_{x \to a}f(x)\) exists
\(\lim_{x \to a}f(x) = f(a)\)
Types of discontinuities:
Removable (hole): \[\lim_{x \to a}f(x) \text{ exists but } \neq f(a)\]
Jump: \[\lim_{x \to a^-}f(x) \neq \lim_{x \to a^+}f(x)\]
Infinite: \[\lim_{x \to a}f(x) = \pm\infty\]
Test your understanding of all limit concepts.