Limits and Continuity

<p>Mathematical analysis is a branch of mathematics that deals with limits, continuity, and infinite processes. One of the fundamental topics in this field is the concept of limits. In this post, we'll explore the notion of a limit, including the formal definition and some common examples.</p> <h3 id="what-is-a-limit">What is a Limit?</h3> <p>In simple terms, the limit describes the behavior of a function as its input approaches a certain value. The formal definition is given as:</p> <p>$$\lim_{{x o a}} f(x) = L$$</p> <p>This expression means that as $(x)$ approaches the value $(a)$, the function $(f(x))$ approaches the value $(L)$. In more rigorous terms, we say that for every $( \epsilon &gt; 0 )$, there exists a $( \delta &gt; 0 )$ such that if $( |x - a| &lt; \delta )$, then $( |f(x) - L| &lt; \epsilon )$.</p> <h3 id="an-example-of-a-limit">An Example of a Limit</h3> <p>Let's consider the following example. Suppose we have the function $( f(x) = 2x + 3 )$. We want to calculate the limit of this function as $( x )$ approaches 1:</p> <p>$$\lim_{{x o 1}} (2x + 3) = 5$$</p> <p>This is an example of a simple linear function. The value of the function at $( x = 1 )$ is exactly 5, so the limit is 5.</p> <h3 id="continuity-of-a-function">Continuity of a Function</h3> <p>A function is said to be continuous at a point $( x = a )$ if the following three conditions hold:</p> <ol> <li>The function is defined at $( x = a )$.</li> <li>The limit of the function as $( x )$ approaches $( a )$ exists.</li> <li>The limit of the function as $( x )$ approaches $( a )$ equals the function's value at $( a )$, i.e., $( \lim_{{x o a}} f(x) = f(a) )$.</li> </ol> <p>If any of these conditions are not met, the function is not continuous at $( x = a )$. For example, the function $( f(x) = rac{1}{x} )$ is not continuous at $( x = 0 )$ because the limit does not exist at that point.</p>