Summary of "1 - Úvod do limity funkce (MAT - Limita a spojitost funkce)"

Overview

This document summarizes the main ideas, examples, and method shown for understanding limits of functions as x approaches a point.

Main ideas / concepts

A limit describes the behavior of a function as x approaches a given point — it inspects the neighborhood around that point (from the left and right), not necessarily the function’s actual value at the point.

Limits are useful at points that are inaccessible by direct substitution (holes, undefined points) because they describe what function outputs approach.
A limit at a point exists only if the left-hand and right-hand approaches agree.
The limit can differ from the function’s actual value at the point (or the function value may be undefined).
Graphs give intuition about left/right behavior; algebraic methods let you compute limits exactly.

Examples

f(x) = (x^2 − 1)/(x − 1)
- Algebraic simplification: x^2 − 1 = (x − 1)(x + 1), so for x ≠ 1, f(x) simplifies to x + 1.
- Graphically: the graph is the line y = x + 1 with a hole at x = 1 (function undefined there).
- Limit: as x → 1 (from left and right), f(x) → 2, so lim_{x→1} (x^2 − 1)/(x − 1) = 2, even though f(1) is undefined.
- Interpretation: a removable discontinuity (hole) where the limit gives the value that would “make sense” at the point.
g(x) = sgn(x) and h(x) = |sgn(x)|
- sgn(x) = −1 for x < 0, 0 at x = 0, +1 for x > 0.
- |sgn(x)| = 1 for x ≠ 0 and 0 at x = 0.
- Function value: h(0) = 0.
- Limit: as x → 0 from either side, h(x) → 1, so lim_{x→0} |sgn(x)| = 1.
- Interpretation: the limit exists and differs from the function’s value at the point.
p(x) = x + 2
- Domain: all real numbers; no problematic points.
- Limits equal function values: e.g., lim_{x→0} (x + 2) = 2 = p(0); lim_{x→−2} (x + 2) = 0 = p(−2).
- Interpretation: for continuous points, the limit equals the function value.

Method — step-by-step (for rational expressions giving an indeterminate form)

To compute a limit for a rational expression that yields an indeterminate form at a point:

Factor numerator and/or denominator if possible (example: x^2 − 1 = (x − 1)(x + 1)).
Cancel the common factor that causes the undefined point (here cancel (x − 1) for x ≠ 1).
Evaluate the simplified expression at the limiting x-value (plug in x = 1 into x + 1 to get 2).
Conclude the limit: lim_{x→1} (x^2 − 1)/(x − 1) = 2.

Practical advice:

Use graphs initially to get intuition about left- and right-hand behavior.
Rely on algebraic simplification to compute limits exactly.
Always check left-hand and right-hand approaches; the limit exists only if they match.

Key takeaways

A limit is about neighborhood behavior, not necessarily the function’s actual value at the point.
Limits can assign a meaningful value at a removable discontinuity (hole).
At continuous points, limit = function value.
A common algebraic tactic: factor and cancel to remove removable singularities, then substitute.