# Limits

Typesetting long calculations involving limits quickly become tedious because the same limit expression (such as "$$\lim_{i \to \infty}$$") appears in each in each step of the calculation—sometimes multiple times!—until the limit is fully evaluated. Any time an expression appears repeatedly, you should consider introducing a macro. In particular, when the input variable for a limit is written as $$i$$, $$j$$, or $$k$$, then the variable almost always is integer index that goes to $$\infty$$. Thus, we introduce macros to abbreviate the corresponding limit expressions. Similarly, $$h$$ is commonly used as an distance that goes to zero, such as in the definition of the derivative, so we define a macro to insert "$$\lim_{h \to 0^+}$$."

Definition

### Examples

Code Output
\ilim 1/i = 0
$$\lim_{i \to \infty} 1/i = 0$$
\jlim 1/j = 0
$$\lim_{j \to \infty} 1/j = 0$$
\klim 1/k = 0
$$\lim_{k \to \infty} 1/k = 0$$
\hlim \frac{f(x + h) - f(x)}{h} = 0
$$\lim_{h \to 0^+} \frac{f(x + h) - f(x)}{h} = 0$$

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