https://calculus.subwiki.org/w/index.php?title=Uniqueness_theorem_for_limits&feed=atom&action=history Uniqueness theorem for limits - Revision history 2021-08-04T08:24:14Z Revision history for this page on the wiki MediaWiki 1.29.2 https://calculus.subwiki.org/w/index.php?title=Uniqueness_theorem_for_limits&diff=548&oldid=prev Vipul: Created page with "==Statement== ===Two-sided limit=== Suppose $f$ is a function and $c$ is a point such that $f$ is defined on both the immediate left and th..." 2011-10-20T20:51:38Z <p>Created page with &quot;==Statement== ===Two-sided limit=== Suppose &lt;math&gt;f&lt;/math&gt; is a <a href="/wiki/Function" title="Function">function</a> and &lt;math&gt;c&lt;/math&gt; is a point such that &lt;math&gt;f&lt;/math&gt; is defined on both the immediate left and th...&quot;</p> <p><b>New page</b></p><div>==Statement==<br /> <br /> ===Two-sided limit===<br /> <br /> Suppose &lt;math&gt;f&lt;/math&gt; is a [[function]] and &lt;math&gt;c&lt;/math&gt; is a point such that &lt;math&gt;f&lt;/math&gt; is defined on both the immediate left and the immediate right of &lt;math&gt;c&lt;/math&gt;. The '''uniqueness theorem for limits''' states that ''if'' the [[fact about::limit]] of &lt;math&gt;f&lt;/math&gt; exists at &lt;math&gt;c&lt;/math&gt; (in the sense of existence as a finite real number) then it is unique. In other words:<br /> <br /> If &lt;math&gt;\lim_{x \to c} f(x) = L&lt;/math&gt; and &lt;math&gt;\lim_{x \to c} f(x) = M&lt;/math&gt;, then &lt;math&gt;L = M&lt;/math&gt;.<br /> <br /> ===Left hand limit===<br /> <br /> Suppose &lt;math&gt;f&lt;/math&gt; is a [[function]] and &lt;math&gt;c&lt;/math&gt; is a point such that &lt;math&gt;f&lt;/math&gt; is defined on the immediate left of &lt;math&gt;c&lt;/math&gt;. The '''uniqueness theorem for left hand limits''' states that ''if'' the left hand limit of &lt;math&gt;f&lt;/math&gt; exists at &lt;math&gt;c&lt;/math&gt; (in the sense of existence as a finite real number) then it is unique. In other words:<br /> <br /> If &lt;math&gt;\lim_{x \to c^-} f(x) = L&lt;/math&gt; and &lt;math&gt;\lim_{x \to c^-} f(x) = M&lt;/math&gt;, then &lt;math&gt;L = M&lt;/math&gt;.<br /> <br /> ===Right hand limit===<br /> <br /> Suppose &lt;math&gt;f&lt;/math&gt; is a [[function]] and &lt;math&gt;c&lt;/math&gt; is a point such that &lt;math&gt;f&lt;/math&gt; is defined on the immediate right of &lt;math&gt;c&lt;/math&gt;. The '''uniqueness theorem for right hand limits''' states that ''if'' the right hand limit of &lt;math&gt;f&lt;/math&gt; exists at &lt;math&gt;c&lt;/math&gt; (in the sense of existence as a finite real number) then it is unique. In other words:<br /> <br /> If &lt;math&gt;\lim_{x \to c^+} f(x) = L&lt;/math&gt; and &lt;math&gt;\lim_{x \to c^+} f(x) = M&lt;/math&gt;, then &lt;math&gt;L = M&lt;/math&gt;.</div> Vipul