Module 6  Limit as x Approaches a  
Introduction  Lesson 1  Lesson 2  Lesson 3  SelfTest  
Lesson 6.3: Limits Graphically, Numerically, and Algebraically  
In this lesson you will use graphical and numerical methods to support algebraic conclusions about limits. The limit of a function may be estimated from a graph of the function or numerically from a table of function values. The Graphical Approach Estimate graphically by using the Trace feature, as described below.
The fact that there is no value beside the ycoordinate shows that Y_{1} is undefined at x = 0. However, the limit as x approaches 0 of the function is defined because you can get as close to 1 as you want by letting x be close enough to 0. The limit can be illustrated by using the Trace feature.
The graph and displayed values provide evidence that when x is close to 0, is close to 1, or Numerical Approach The Table feature of the TI83 can be used to evaluate for values of x close to 0.
In the ASK mode you can enter any value you choose for x in the table and the calculator will generate the corresponding y value for functions selected in the Y= editor.
The table provides numerical evidence that .
The Sandwich Theorem The Sandwich theorem is used in many calculus books to prove that . Consider the following argument:
This argument is called the Sandwich Theorem because the values of sin(x)/x are "sandwiched" between the values of cos(x) and 1. Simultaneous Graphing The Sandwich theorem can be illustrated by graphing the three terms of the compound inequality simultaneously.
The simultaneous convergence of three graphs to 1 around x = 0 illustrates the Sandwich theorem's argument that . Redrawing the Graphs You can see the three functions converge again using the Clear Draw command.
Left and RightHand Limits The notation represents a righthand limit and it is read "the limit of as x approaches 0 from the right." It represents the value approached by as x approaches 0 through positive values, which are to the right of 0, as shown below. The diagram illustrates the xvalues approaching 0 through positive values and the corresponding values of . The notation represents a lefthand limit and it is read "the limit of as x approaches 0 from the left." It represents the value approached by as x approaches 0 through negative values, which are to the left of 0, as shown below. The diagram illustrates the xvalues approaching 0 through negative values and the corresponding values of . Using the Trace Feature The Trace feature may be used to estimate these limits. As the Trace cursor moves along a curve, the x and yvalues appear at the bottom of the Graph screen. Look at the yvalues of as x approaches 0 from the left and from the right by using the Trace feature.
Moving the Trace cursor toward zero from the left and from the right provides graphical and numeric reinforcement for the left and righthand limits shown below. Entering a Specific xValue While the Trace cursor is active, you can move it to a particular point by typing in the xcoordinate of the desired point. For example, move to the point with x = 0.001 while the Trace cursor is active.
The cursor is not visible because it is below the viewing window, but the coordinates of the cursor are shown giving further evidence that . 

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