fix descriptions in 1.3 and 1.4

ptx/sec_limit_analytically.ptx

@@ -483,19 +483,19 @@
     <image width="47%">
       <description>
         <p>
-          Graph illustrating the squeeze theorem. There are three functions, <m>h(x)</m>,
+          An illustration of the squeeze theorem. There are graphs of three functions shown, labelled <m>h(x)</m>,
           <m>g(x)</m>, and <m>f(x)</m>. On the <m>y</m>axis, there is a marker at <m>y = 4</m>,
           labeled <m>L</m> and on the <m>x</m>axis there is a marker at <m>x = 5</m>, labeled
           <m>c</m>.
         </p>
         <p>
           For all values of <m>x \leq c</m> <m>f(x) \leq L</m> and <m>h(x) \geq L</m>.
-          For all values of <m>x</m> <m>f(x) \leq g(x) \leq h(x)</m>, that is, the line
-          of the function <m>g(x)</m> is between the lines of the functions <m>g(x)</m>
+          For all values of <m>x</m> <m>f(x) \leq g(x) \leq h(x)</m>, that is, the graph
+          of the function <m>g(x)</m> lies between the graphs of the functions <m>g(x)</m>
           and <m>f(x)</m>.
         </p>
         <p>
-          The graph shows that where <m>x = c</m>, <m>f(x)</m> and <m>h(x)</m> converge
+          The image shows that where <m>x = c</m>, <m>f(x)</m> and <m>h(x)</m> converge
           on <m>y = L = 4</m>. Because <m>f(x) \leq g(x) \leq h(x)</m>, we can extrapolate
           that <m>\lim\limits_{x\to \c} g(x) = L</m> too.
         </p>
@@ -953,15 +953,15 @@
         <image width="47%">
           <description>
             <p>
-              Graph of the linear equation <m>(x^1-1)/(x-1)</m>, shows the function with
-              the <m>y</m> interval <m>0</m> to <m>3</m> and <m>x</m> interval <m>0</m>
-              to <m>2</m>. The function has a <m>y</m> intercept at <m>y = 1</m>, and
-              is undefined at <m>x = 1</m>. The graph has a positive slope.
+              Graph of the function <m>(x^2-1)/(x-1)</m>, showing the region with
+              <m>y</m> from <m>0</m> to <m>3</m> and <m>x</m> from <m>0</m>
+              to <m>2</m>. The graph is a straight line, with slope 1 and a <m>y</m> intercept at <m>y = 1</m>,
+              except for a hole at the point <m>(1,2)</m>, since the function is undefined at <m>x = 1</m>.
             </p>
           </description>
           <shortdescription>
             Graph of the polynomial x squared minus 1 divided by the polynomial x minus 1.
-            Is undefined at x = 1.
+            It is the same as the line y=x+1, except that it is undefined at x = 1.
           </shortdescription>
           <latex-image label="img_limitxplus1">
             \begin{tikzpicture}

ptx/sec_limit_onesided.ptx

@@ -183,30 +183,31 @@
             <p>
               Graph of the piecewise function
               <m>f(x) = \begin{cases} x \amp 0\leq x\leq 1 \\ 3-x \amp 1\lt x\lt 2 \end{cases}</m>.
-              There are two lines,
-              one with a positive slope and the other with a negative slope.
+              There are two line segments: for <m>0\leq x\leq 1</m> we have a line segment
+              with positive slope, and for <m>1\lt x\lt 2</m> we have a line segment with negative slope.
             </p>
             <p>
-              The line with a positive slope starts at the point <m>(0, 0)</m>
-              and ends at <m>(1, 1)</m>. The line with a negative slope starts
+              The line segment with a positive slope starts at the point <m>(0, 0)</m>
+              and ends at <m>(1, 1)</m>. The line segment with a negative slope starts
               at <m>(1, 2)</m> and ends at <m>(2, 1)</m>.
             </p>
             <p>
-              The start and end points of the line with a positive slope
-              are solid dots, indicating that the function is defined
-              at those points. The start and end points of the line
-              with a negative slope are hollow dots. This tells us that
-              the function is undefined for <m>x = 1</m> when going right
-              to left and <m>x = 2</m> is undedined as well. Because the
-              function is defined at <m>x = 1</m> on the line with a positive
-              slope, but is undefined at <m>x = 1</m> on the other line, we can
-              tell the function <m>f</m> has a one-sided limit as <m>x</m>
-              approaches <m>1</m>.
+              The start and end points of the line segment with a positive slope
+              are solid dots, indicating that those points are part of the graph. 
+              The start and end points of the line segment
+              with a negative slope are hollow dots. 
+              This tells that although the second line segment gets arbitrarily close
+              to the points <m>(1,2)</m> and <m>(2,1)</m>, these points are not part of the graph.
+            </p>
+
+            <p>
+              Since <m>f(x)</m> is close to <m>1</m> when <m>x</m> is close to 1, but <m>x\lt 1</m>,
+              while <m>f(x)</m> is close to <m>2</m> when <m>x</m> is close to 1, but <m>x\gt 1</m>,
+              we can conclude that the left and right hand limits are different.
             </p>
           </description>
           <shortdescription>
-            Graph of a piecewise function that has only a left sided limit as x
-            approaches 1.
+            Graph of a piecewise function that has different left and right hand limits when x=1.
           </shortdescription>
           <latex-image label="img_onesided1">
             \begin{tikzpicture}
@@ -532,23 +533,22 @@
           <description>
             <p>
               Graph of <m>f</m> from <xref ref="ex_onesideb"/>.
-              There are
-              two lines shown,
-              the first starts at the point <m>0, 2</m> and ends at
-              <m>(1, 1)</m>. The second start at <m>(1, 1)</m> and ends
-              at <m>(2, 0)</m>. Both lines have a negative slope and are
-              undefined at their start and end points, which are marked with
-              hollow dots. The first line is straight, but the second has a
-              small upward curve to it.
+              The graph consists of two parts.
+              The first part is a line that starts at the point <m>(0, 2)</m> and ends at
+              <m>(1, 1)</m>. The second part is a curve that starts at <m>(1, 1)</m> and ends
+              at <m>(2, 0)</m>.
+              The points <m>(0,2)</m>, <m>(1,1)</m>, and <m>(2,0)</m> are all marked with hollow dots,
+              indicating that although the graph gets close to these points, they are not part of the graph.
             </p>
             <p>
-              The graph may be undefined for <m>x = 1</m>, but since
-              the equation has a left and right limit, there for
-              <m>f(x)</m> has a limit as <m>x</m> approaches <m>1</m>.
+              The function is undefined for <m>x = 1</m>,
+              but the graph shows that <m>f(x)</m> approaches the same value (namely, 1)
+              from both the left and the right, allowing us to conclude that <m>\lim_{x\to 1}f(x)</m> exists,
+              and is equal to 1.
             </p>
           </description>
           <shortdescription>
-            Graph of the piecewise function from the previous example.
+            Graph of a piecewise-defined function. It is undefined when x=1, but has a limit at this point.
           </shortdescription>
           <latex-image label="img_onesidedb">
             \begin{tikzpicture}[declare function = {func(\x) = (\x &lt; 1) * (2 - x) + (\x &gt; = 1) * ((x - 2)^2);}]