Question

X стрем 0 lim ln (1+2x)/5x

Answer 1

Решение в приложенном файле.

Answer 2

Решение на фотографии

Answer 3

To find the limit of the expression as x approaches 0, we can use L'Hôpital's rule, which states that for certain indeterminate forms, the limit of the ratio of two functions is equal to the limit of the ratio of their derivatives. The indeterminate form in this case is 0/0, so we can apply L'Hôpital's rule.

Let's differentiate the numerator and the denominator with respect to x and then take the limit as x approaches 0.

Given the expression:

lim(x → 0) [(ln(1 + 2x)) / (5x)]

Step 1: Differentiate the numerator and denominator with respect to x.

The derivative of ln(1 + 2x) with respect to x is:

d/dx [ln(1 + 2x)] = 2 / (1 + 2x)

The derivative of 5x with respect to x is:

d/dx [5x] = 5

Step 2: Rewrite the limit after differentiating the numerator and denominator.

lim(x → 0) [(2 / (1 + 2x)) / 5]

Step 3: Now, take the limit as x approaches 0.

lim(x → 0) [(2 / (1 + 2x)) / 5] = (2 / (1 + 2 * 0)) / 5 = 2 / 5

Therefore, the limit of the given expression as x approaches 0 is 2/5.