Introduction

This section of the unit Analytical/Algebraic Limits and their properties. Graphs and tables are cool and useful tools, but now applying limits to functions will allow us to perform more complex analytical operations on the functions.

How to do limits analytically/algebraically

Just plug in the value the independent variable (usually $x$) is approaching into the function you are taking the limit of.

Example 1:

Calculate $\underset{x\to 2}{\lim }2x^2-5x+3$

$\underset{x\to 2}{\lim }2x^2-5x+3=2(2{)}^2-5(2)+3=1$

That’s it!

The concept of a one-sided limit can be extended here as well.

Example 2:

Calculate $\underset{x\to 2^{+}}{\lim }\ln (\mathrm{x}+2)$

Despite asymptotic behavior we can still calculate the limit. Recall that the natural log of 0 is undefined, but the natural log of very small positive numbers are negative numbers that are large in magnitude. So from what we know from limits we can reason this one-sided limit is $-\infty$. 

Just like with other representations of limits, if the left side and right side limits don’t equal each other, the limit does not exist.

Finally, there are some properties of limits to know:

Limits can be added, subtracted, multiplied, divided, or be part of a composite function to evaluate the limits of sums, differences, etc using the composite limit theorem.

In notation:

If $\underset{x\to c}{\lim }f(x)=M$ and $\underset{x\to c}{\lim }g(x)=N$, then

$\underset{x\to c}{\lim }[f(x)+g(x)]=M+N$

$\underset{x\to c}{\lim }[f(x)-g(x)]=M-N$

$\underset{x\to c}{\lim }[f(x)\cdot g(x)]=MN$

$\underset{x\to c}{\lim }[\frac{f(x)}{g(x)}]=\frac{M}{N},N\ne 0$

If $\underset{x\to c}{\lim }g(x)=L$ and $\underset{x\to L}{\lim }f(x)=f(L)$ then

$\underset{x\to c}{\lim }f(g(x))=f(\underset{x\to c}{\lim }g(x))=f(L)$

Note the differences in the composite limit theorem from the rest. The outer function is approaching the value of the limit from the inner function.

Example 3:

If ${\lim }_{x\to 4}f(x)=3$ and $g(x)=x+5$, calculate $\underset{x\to 4}{\lim }g(f(x))$.

The conditions of the composite limit theorem hold up, so we can apply it.

$\underset{x\to 4}{\lim }g(f(x))=g(\underset{x\to 4}{\lim }f(x))=g(3)=8$

Practice

Independently practice the limit strategies you learned in this article!