# barcodelib.barcode.asp.net.dll free download >> limit(x + 5,3) ans = 8 in .NET framework Creator Denso QR Bar Code in .NET framework >> limit(x + 5,3) ans = 8

>> limit(x + 5,3) ans = 8
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EXAMPLE 6-1 Let f ( x ) = 2 x +21 and g(x) = x2 + 1 Compute the limit as x 3 of both functions and x verify the basic properties of limits using these two functions and MATLAB SOLUTION 6-1 First we tell MATLAB what symbolic variables we will use and define the functions:
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>> syms x >> f = (2*x + 1)/(x 2); >> g = x^2 + 1;
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Now let s find the limit of each function, and store the result in a variable we can use later:
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>> F1 = limit(f,3) F1 = 7
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>> F2 = limit(g,3) F2 = 10
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CHAPTER 6 Symbolic Calculus Differential Eqs
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The first property of limits we wish to verify is: lim( f ( x ) + g( x )) = lim f ( x ) + lim g( x ) x a x a x a From our calculations so far we see that: lim f ( x ) + lim g( x ) = 7 + 10 = 17 x 3 x 3 Now let s verify the relation by calculating the left-hand side:
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>> limit(f+g,3) ans = 17
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Next we can verify: lim k f ( x ) = k lim f ( x ) x a x a for any constant k Let s let k = 3 for which we should find: lim k f ( x ) = k lim f ( x ) = (3)( 7) = 21: x a x a
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>> k=3; >> limit(k*f,3) ans = 21
Now let s check the fact that the limit of the product of two functions is the product of their limits, that is: lim f ( x ) g( x ) = lim f ( x ) lim g( x ) x a x a x a The product of the limits is:
>> F1*F2 ans = 70
And we find the limit of the product to be:
>> limit(f*g,3) ans = 70
MATLAB Demysti ed
Finally, let s verify that: lim f ( x ) g ( x ) = lim f ( x ) x a x a We can create f (x)g(x) in MATLAB:
>> h = f^g h = ((2*x+1)/(x 2))^(x^2+1)
lim g ( x )
x a
Computing the limit:
>> limit(h,3) ans = 282475249
Checking the right side of the relation, we find that they are equal:
>> A = F1^F2 A = 282475249
As an aside, we can check if two quantities in MATLAB are equal by calling the isequal command If two quantities are not equal, isequal returns 0 Recall that earlier we defined a constant k = 3 Here is what MATLAB returns if we compare it to A = F1^F2:
>> isequal(A,k) ans = 0
CHAPTER 6 Symbolic Calculus Differential Eqs
On the other hand:
>> isequal(A,limit(h,3)) ans = 1
Computing lim f ( x ) x
limit(f,inf)
We can calculate limits of the form lim f ( x ) by using the syntax: x Let s use MATLAB to show that lim
x
1 x2 + x x = : 2
>> limit(sqrt(x^2+x) x,inf) ans = 1/2
We can also calculate lim f ( x ) For example:
x
>> limit((5*x^3 + 2*x)/(x^10 + x + 7), inf) ans = 0
MATLAB will also tell us if the result of a limit is For example, we verify that lim 1 = : x 0
>> limit(1/abs(x)) ans = Inf
LEFT- AND RIGHT-SIDED LIMITS
When a function has a discontinuity, the limit does not exist at that point To handle limits in the case of a discontinuity at x = a, we define the notion of left-handed and right-handed limits A left-handed limit is defined as the limit as x a from the left, that is x approaches a for values of x < a In calculus we write:
x a
lim f ( x )
MATLAB Demysti ed
For a right-handed limit, where x a from the right, we consider the case when x approaches a for values of x > a The notation used for right-handed limits is:
x a +
lim f ( x )
If these limits are equal, then lim f ( x ) exists In MATLAB, we can compute leftx a and right-handed limits by passing the character strings left and right to the limit command as the last argument We must also tell MATLAB the variable we are using to compute the limit in this case Let s illustrate with an example EXAMPLE 6-2 Show that lim x 3 does not exist x 3