Math CentralQuandaries & Queries


Question from Mac, a student:

can you please tell me what is the reason they say "denominator is a negative quantity"
in the solution 11 and "denominator is a positive quantity" solution 10 ??

If i guess correctly, for solution 10, its because of x^2 in the denominator.

Thanks for your help.

Hi Mac,

In both of these examples you are taking a limit of a fraction as x approaches zero. In each case the numerator approaches a non-zero constant as x approaches zero and the denominator approaches zero. Hence we might say the limit will approach infinity or the limit doesn't exist, but with more care we can sometimes be more explicit.

In Solution 10 the fraction is (x2 - 7)/x2. Your observation is correct, the denominator is x2 and hence is a positive quantity. (Remember that x can not be zero.) On the other hand when x is close to zero x2 is small, certainly less than 7, and hence the numerator x2 - 7 will be negative. Thus for x close to zero the fraction will be negative. Try a few values with your calculator, x = 0.1, x = -0.1, x = 0.01, x = -0.01, x = 0.001, x = -0.001, ... The value of the fraction is always negative and increases in absolute value as x approaches 0. Thus the limit is -∞.

In Solution 11 the fraction is (x4 + 5x -3)/[2 - √(x2 + 4)]. Again as x approaches zero the numerator approaches a negative value, this time -3. In the denominator x2 is positive and hence x2 + 4 is larger than 4 so √(x2 + 4) is larger than √ 4 = 2. Hence [2 - √(x2 + 4)] is negative. Thus when x is close to zero the faction is a negative quantity divided by a negative quantity and thus positive. Therefore, in this case the limit is +∞.

I hope this helps,

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