Doug C. answered • 12/31/25
Math Tutor with Reputation to make difficult concepts understandable
The key to understanding this is that the expression c/0 (where c is a constant) is undefined, but if the denominator gets close to zero (but not equal to zero) the expression approaches either ±∞. The key is to determine which it is (positive or negative).
In this particular case let's see what happens as x -> 5 (in the post the minus sign indicates getting close to 5 from the left (so values like 4.9, 4.99, 4.999...
(x+1)/(x - 5) as x -> 5 the numerator gets close to 6 which is positive.
The denominator gets close to zero, but with either positive or negative values depending on which side of 5. With values like 4.9 the denominator is negative: (4.9 - 5) = -0.1.
So think +/-0 is negative, so (x+1)/(x-5) -> −≈ as x -> 5-.
If the limit was as x -> 5+, the numerator would still be positive, but the denominator (5.1 - 5) = 0.1 > 0, i.e. positive. Think +/+0, +∞.
Visit this graph to see a table with values of x close to 5 and the corresponding values for (x+1)/(x - 5).
desmos.com/calculator/a7dmcokyfl
