Rules for limits

When working with limits, it can quickly become complicated to calculate them directly from the definition. Fortunately, there are a number of simple calculation rules that make it possible to find many limits without long derivations. These rules resemble the usual arithmetic rules known from algebra.

 

 

Sum and difference

If two functions have a limit at the same point, their limits can be added or subtracted:

 

$$ \large \lim_{x \to a} [f(x) \pm g(x)] = \lim_{x \to a} f(x) \pm \lim_{x \to a} g(x) $$

 

This means that each limit can be calculated separately, and then the addition or subtraction is applied to the results.

 

 

Product

The limit of a product is the product of the limits:

 

$$ \large \lim_{x \to a} [f(x) \cdot g(x)] = \left( \lim_{x \to a} f(x) \right) \cdot \left( \lim_{x \to a} g(x) \right) $$

 

Thus, the two limits are multiplied together, provided that both exist and are finite.

 

 

Quotient

For a fraction, the limit can be found by taking the limit of numerator and denominator separately, as long as the denominator’s limit is not zero:

 

$$ \large \lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)} \quad \text{if } \lim_{x \to a} g(x) \neq 0 $$

 

If the denominator’s limit is zero, the expression must be examined more closely, since infinite limits or 0/0 situations may occur, which require rewriting or factorization.

 

 

Constant times function

A constant can always be taken outside the limit sign:

 

$$ \large \lim_{x \to a} [k \cdot f(x)] = k \cdot \lim_{x \to a} f(x) $$

 

This makes many expressions easier to handle, since factors that do not depend on x can be ignored.

 

 

Powers and roots

If the function has a limit, powers and roots can be applied directly to the result:

 

$$ \large \lim_{x \to a} [f(x)]^n = \left( \lim_{x \to a} f(x) \right)^n $$

$$ \large \lim_{x \to a} \sqrt[n]{f(x)} = \sqrt[n]{\lim_{x \to a} f(x)} $$

 

These rules apply as long as the resulting power or root is defined (for example, one cannot take the square root of a negative number in the real numbers).

 

 

Composite functions

If a function consists of another function, the limit can be found by first taking the limit of the inner function and then substituting the result into the outer one:

 

$$ \large \lim_{x \to a} f(g(x)) = f\!\left( \lim_{x \to a} g(x) \right) $$

 

This holds only if f is continuous at the point approached by g(x).

 

 

Examples

1. Calculate the limit:

 

$$ \large \lim_{x \to 2} (3x^2 - 5x + 4) $$

 

Here we can simply substitute x = 2, since polynomials are continuous:

 

$$ \large 3 \cdot 2^2 - 5 \cdot 2 + 4 = 12 - 10 + 4 = 6 $$

 

2. Calculate the limit:

$$ \large \lim_{x \to 3} \frac{x^2 - 9}{x - 3} $$

 

Direct substitution gives 0/0, so we rewrite the expression by factoring:

 

$$ \large \frac{x^2 - 9}{x - 3} = \frac{(x - 3)(x + 3)}{x - 3} = x + 3 $$

 

and therefore

 

$$ \large \lim_{x \to 3} \frac{x^2 - 9}{x - 3} = 6 $$

 

 

In this way, the rules can be used to simplify and calculate limits efficiently.

 

 

Summary

The rules for limits make it possible to work with limits almost like with ordinary numbers. They apply to all functions where the limits exist and form the basis for most calculations in analysis.