Question

When x=2e lim h →0 (ln (x+h)-ln (x))/h is

A. 1/2e
B. 1
C. ln (2e)
D. nonexistent

Answer 1

To find the limit of the given expression, we can simplify it and apply limit properties. The final limit is 1.

Step-by-step explanation:

To find the limit of the expression lim h →0 (ln (x+h)-ln (x))/h, we can use the fact that the natural logarithm is the inverse of the exponential function. Let's apply the property that ln(1 + p) ≈ p for small values of p. Since x = 2e, we have ln(x) = ln(2e) = ln(2) + ln(e) = ln(2) + 1, as ln(e) = 1. Plugging these values into the expression, we get:

lim h →0 (ln(2+h) - (ln(2) + 1))/(h)

Now we can simplify further:

= lim h →0 (ln(2) + ln(1+h) - ln(2) - 1)/(h)

= lim h →0 (ln(1+h) - 1)/(h)

By applying the limit properties, we can take the limit of the numerator and denominator separately:

= ln'(1) = 1/1 = 1

Therefore, the limit of the given expression is B. 1.