Limits of functions

Japanese school year: Math III

What you learn

This topic explores the limits and continuity of functions as the input variable approaches a specific value or infinity. It forms the indispensable foundation for defining derivatives and finding graph asymptotes in calculus. Prior fluency in graphing functions, factoring polynomials, and algebraic manipulation is recommended.

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Key points

This formula shows that the ratio of $\sin x$ to $x$ approaches 1 as angle $x$ nears 0. It is essential for deriving the derivatives of trigonometric functions, where $x$ must be measured in radians.

\lim_{x \to 0} \frac{\sin x}{x} = 1

This formula defines the mathematical constant $e$ (Euler's number, about 2.718). Adding a tiny number $x$ to 1 and raising it to the power of $\frac{1}{x}$ converges to $e$ as $x$ approaches 0.

\lim_{x \to 0} (1+x)^{\frac{1}{x}} = e

This equation defines that a function $f(x)$ is continuous (unbroken) at $x = a$ . It means the value that $f(x)$ approaches as $x$ nears $a$ matches the actual value $f(a)$ at that point.

\lim_{x \to a} f(x) = f(a)

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Practice 1Coming soon