2.2 HW Problems

Table of Contents:

An Informal Description of the Limit of a Function

• Suppose that f(x) is defined on an open interval about c, except possibly at c itself (meaning it never reaches c).
• If f(x) is arbitrarily close to the number L for all x sufficiently close to c, but NOT c, then we say that f approaches the limit L as x approaches c, and write:

$$ \lim_{x \, \rightarrow \, c} f(x)=L $$

This reads as “the limit f(x) as x approaches c is L”.

$$ \lim_{x \, \rightarrow \, 1} f(x)=2 \enspace or \enspace \lim_{x \, \rightarrow \, 1} \frac{x^2-1}{x-1}=2 $$

• The definition says that the values of f(x) are CLOSE to the number L whenever x is close to c.

The limits of f(x), g(x), and h(x) all equal 2 as x approaches 1. However, only h(x) has the same function value as its limit at $x =1$.

Finding the Limit

• We find the limits of the identity function and of a constant function as x approaches $x = c$.

  • (a) If f is the identity function $f(x)=x$, then for any value of c:

  $$ \lim_{x \, \rightarrow \, c} f(x)= \lim_{x \, \rightarrow \, c} x = c $$

  

  • (b) If f is the constant function $f(x)=k$ (function with the constant value k), then for any value c:

    $$ \lim_{x \, \rightarrow \, c} f(x)= \lim_{x \, \rightarrow \, c} k = k $$

  $$ \lim_{x \, \rightarrow \, 3} x = 3 \quad \textrm{Limit of identity function at x = 3} $$

• A function may not have a limit at a particular point. Some ways that limits can fail to exist are illustrated below:

The Limit Laws

• A few basic rules allow us to break down complicated functions into simple ones when calculating limits.

Theorem 1 — Limit laws