Limits and Derivatives - Limits

Limit of a Function: The limit of a function $f(x)$ as $x$ approaches $a$ is the value $L$ that $f(x)$ gets closer to as $x$ moves arbitrarily close to $a$ from both sides. Visually, imagine tracing the graph of $f(x)$ with your finger from both the left and right toward the x-coordinate $a$; the height (y-value) your finger approaches is the limit, denoted as $\lim_{x \to a} f(x) = L$.

Left-Hand and Right-Hand Limits: A limit exists at $x = a$ if and only if the Left-Hand Limit (LHL), $\lim_{x \to a^-} f(x)$, and the Right-Hand Limit (RHL), $\lim_{x \to a^+} f(x)$, are equal. On a graph, this means the two ends of the curve must meet at the same point, even if there is a 'hole' exactly at $x = a$.

Indeterminate Forms: If substituting $x = a$ into the function results in an expression like $\frac{0}{0}$ or $\frac{\infty}{\infty}$, it is called an indeterminate form. Visually, this often indicates a 'removable discontinuity' where the function is undefined at a single point but the surrounding curve remains continuous and points toward a specific value.

Algebra of Limits: Limits distribute over basic operations. The limit of a sum is the sum of the limits, and the same applies to subtraction, multiplication, and division (provided the denominator's limit is not zero). This allows us to break down complex expressions into simpler parts to evaluate them step-by-step.

Sandwich (Squeeze) Theorem: If we have three functions such that $g(x) \le f(x) \le h(x)$ for all $x$ near $a$, and both $g(x)$ and $h(x)$ approach the same limit $L$ at $a$, then $f(x)$ must also approach $L$. Visually, the graph of $f(x)$ is 'trapped' or 'squeezed' between the other two graphs, forcing it to pass through the same point at $x = a$.

Evaluation of Rational Limits: For rational functions, if direct substitution leads to $\frac{0}{0}$, we use algebraic techniques like factorization, rationalization of the numerator/denominator, or standard identities to simplify the expression and remove the factor causing the zero before re-evaluating.

Trigonometric Limits: A fundamental concept is $\lim_{x \to 0} \frac{\sin x}{x} = 1$. Visually, this occurs because for very small values of $x$ (measured in radians), the length of the vertical segment $\sin x$ in a unit circle becomes almost identical to the length of the arc $x$, making their ratio approach 1.