Formal Definition & Notation

A limit represents the value that a function approaches as the input gets arbitrarily close to a point.

This means that as $x$ gets arbitrarily close to $a$ where approaches $L$.

One-Sided & Two-Sided Limits

A limit can be approached from either side:

• Left-hand limit: approaching $a$ from values smaller than $a$: $$\lim_{x \rarr a^-}f(x)$$
• Right-hand limit: approaching $a$ from values larger than $a$: $$\lim_{x \rarr a^+}f(x)$$
• The limit exists only if both one-sided limits are equal: $$\lim_{x \rarr a^-}f(x) = \lim_{x \rarr a^+}f(x)$$

When Limits Fail to Exist

A limit does not exist in the following cases:

• Jump discontinuity: $$\lim_{x \rarr a^-}f(x) \neq \lim_{x \rarr a^+}f(x)$$
  • Example: a step function where the left and right limits are different.
• Infinite limit: $$\lim_{x \rarr 0}\frac{1}{x^2}=\infty$$
  • The function grows unbounded.
• Oscillation: $$\lim_{x \rarr 0}\sin\left(\frac{1}{x}\right)$$
  • The function fluctuates infinitely without settling to a single value.

Special Case – Limits at Infinity

When $x$ approaches infinity, we analyze the end behavior of functions:

• Rational functions: $$\lim_{x \rarr \infty}\frac{1}{x}=0$$
• Polynomial growth: $$\lim_{x \rarr \infty}\frac{x^2}{x}=\infty$$
• Dominant term rule: $$\lim_{x \to \infty} \frac{a x^m}{b x^n} = \begin{cases} 0,\ \text{if } m < n,\\ \frac{a}{b},\ \text{if } m = n, \\ \pm \infty,\ \text{if } m > n. \end{cases}$$