Question

Determine the limit of 4 / x - 1 = 4 / ln x as x approaches a value:

a) 1
b) e
c) 0
d) [infinity]

Answer 1

The limits of the function 4 / x - 4 / ln(x) as x approaches 1 and infinity are 0, as x approaches e is 4/e - 4, and as x approaches 0, the limit does not exist due to divergence.

Step-by-step explanation:

The student's question involves determining the limit of the function 4 / x - 4 / ln(x) as x approaches different values. Specifically, we are looking at the limits as x approaches 1, the mathematical constant e (approximately 2.71828), 0, and ∞ (infinity).

• As x approaches 1, the terms 4/x and 4/ln(x) both approach 4, so the overall limit is 0.
• As x approaches e, ln(e) = 1, so the second term becomes 4. Since the first term also approaches 4/e, the overall limit is 4/e - 4.
• As x approaches 0 from the right, both terms approach infinity, but the first term approaches infinity more rapidly. Hence, the limit does not exist because the function will diverge.
• As x approaches infinity, both terms approach 0. Thus, the limit is 0 for this case.

The exponential function eln(x) = x and its inverse, the natural logarithm ln(ex) = x, are essential to understanding these limits.