12.2 Finding limits: properties of limits

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In this section, you will:

Find the limit of a sum, a difference, and a product.
Find the limit of a polynomial.
Find the limit of a power or a root.
Find the limit of a quotient.

Consider the rational function

f (x) = \frac{x^{2} - 6 x - 7}{x - 7}

The function can be factored as follows:

f (x) = \frac{(x - 7) (x + 1)}{x - 7}, which gives us f (x) = x + 1, x \neq 7.

Does this mean the function $f$ is the same as the function $g (x) = x + 1 ?$

The answer is no. Function $f$ does not have $x = 7$ in its domain, but $g$ does. Graphically, we observe there is a hole in the graph of $f (x)$ at $x = 7,$ as shown in [link] and no such hole in the graph of $g (x),$ as shown in [link] .

Graph of an increasing function where f(x) = (x^2-6x-7)\(x-7) with a discontinuity at (7, 8) — The graph of function $f$ contains a break at *$x = 7$* and is therefore not continuous at $x = 7.$

Graph of an increasing function where g(x) = x+1 — The graph of function $g$ is continuous.

So, do these two different functions also have different limits as $x$ approaches 7?

Not necessarily. Remember, in determining a limit of a function as $x$ approaches $a,$ what matters is whether the output approaches a real number as we get close to $x = a .$ The existence of a limit does not depend on what happens when $x$ equals $a .$

Look again at [link] and [link] . Notice that in both graphs, as $x$ approaches 7, the output values approach 8. This means

\lim_{x \to 7} f (x) = \lim_{x \to 7} g (x) .

Remember that when determining a limit, the concern is what occurs near $x = a,$ not at $x = a .$ In this section, we will use a variety of methods, such as rewriting functions by factoring, to evaluate the limit. These methods will give us formal verification for what we formerly accomplished by intuition.

Finding the limit of a sum, a difference, and a product

Graphing a function or exploring a table of values to determine a limit can be cumbersome and time-consuming. When possible, it is more efficient to use the properties of limits , which is a collection of theorems for finding limits.

Knowing the properties of limits allows us to compute limits directly. We can add, subtract, multiply, and divide the limits of functions as if we were performing the operations on the functions themselves to find the limit of the result. Similarly, we can find the limit of a function raised to a power by raising the limit to that power. We can also find the limit of the root of a function by taking the root of the limit. Using these operations on limits, we can find the limits of more complex functions by finding the limits of their simpler component functions.

A General Note

Properties of limits

Let $a, k, A,$ and $B$ represent real numbers, and $f$ and $g$ be functions, such that $\lim_{x \to a} f (x) = A$ and $\lim_{x \to a} g (x) = B .$ For limits that exist and are finite, the properties of limits are summarized in [link]

Constant, k	$\lim_{x \to a} k = k$
Constant times a function	$\lim_{x \to a} [k \cdot f (x)] = k \lim_{x \to a} f (x) = k A$
Sum of functions	$\lim_{x \to a} [f (x) + g (x)] = \lim_{x \to a} f (x) + \lim_{x \to a} g (x) = A + B$
Difference of functions	$\lim_{x \to a} [f (x) - g (x)] = \lim_{x \to a} f (x) - \lim_{x \to a} g (x) = A - B$
Product of functions	$\lim_{x \to a} [f (x) \cdot g (x)] = \lim_{x \to a} f (x) \cdot \lim_{x \to a} g (x) = A \cdot B$
Quotient of functions	$\lim_{x \to a} \frac{f (x)}{g (x)} = \frac{\lim_{x \to a} f (x)}{\lim_{x \to a} g (x)} = \frac{A}{B}, B \neq 0$
Function raised to an exponent	$\lim_{x \to a} {[f (x)]}^{n} = {[\lim_{x \to \infty} f (x)]}^{n} = A^{n},$ where $n$ is a positive integer
n th root of a function, where n is a positive integer	$\lim_{x \to a} \sqrt[n]{f (x)} = \sqrt[n]{\lim_{x \to a} [f (x)]} = \sqrt[n]{A}$
Polynomial function	$\lim_{x \to a} p (x) = p (a)$

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Source: OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6

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