. Applied calculus; principles and applications . verges to the value 2 is assigned as the value of the function when x = 2. If x = 2 + h, lim^|±4^^^=lim(4 + /i) = 4; h=0 Z -h Al — Z h=Q lim = 4. x=2 X -2 Thus the true or limiting value of this function which takesthe indeterminate form 0/0 is values of x other than 2, y = -—o = ^ + 2; X-4:l X- 2j^=2 = lim {x + 2) = 4. On the graph of y = x -\- 2, the ordinates of points forvalues of x other than 2 represent the values of the function,but for X = 2, the function having no definite value may berepresented by any ordinate lying along the l
. Applied calculus; principles and applications . verges to the value 2 is assigned as the value of the function when x = 2. If x = 2 + h, lim^|±4^^^=lim(4 + /i) = 4; h=0 Z -h Al — Z h=Q lim = 4. x=2 X -2 Thus the true or limiting value of this function which takesthe indeterminate form 0/0 is values of x other than 2, y = -—o = ^ + 2; X-4:l X- 2j^=2 = lim {x + 2) = 4. On the graph of y = x -\- 2, the ordinates of points forvalues of x other than 2 represent the values of the function,but for X = 2, the function having no definite value may berepresented by any ordinate lying along the line x = 2. Ofthe values that may be assigned to the function for x = 2,there is one value represented by MP = 4, which is thelimit of the values represented by the ordinates of points on INDETERMINATE FORMS 427 y = X + 2 SiS X approaches 2; and it is desirable to selectthis value of y as the value of the function when x = 2. By this selection the function is defined for a; = 2 and thusbecomes continuous through that value of the variable In general, lim / (x) defines the value of the function when f{x) is indeterminate for x = a. The expression f {x)]adenotes the value of / (x) when x = a. ; The value of a function of x ior x = a usually means theresult obtained by substituting a for x in the function. When, however, the substitution results in any one of theindeterminate forms. 0/0, 00/OO, , 00-00, 0 00 the definition must be enlarged; thus, the value of a functionfor any particular value of its variable is the limit whichthe function approaches when the variable approaches thisparticular value as its limit. This definition need be used only when the ordinarymethod of getting the value of the function gives rise to anindeterminate form. 428 INTEGRAL CALCULUS 220. Evaluation of Indeterminate Forms. — In many-cases the limits desired are easily found by simple algebraictransformations or by the use of series. When the functionthat assumes the indeterminate form is
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