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,
Penny