Question

Найдите предел функции (sqrt(1+2x) - sqrt(1+x))/(sqrt(9+2x)-sqrt(9-x)).

Answer 1

Ответ:

Пошаговое объяснение:

См фото

Answer 2

Function Limit Calculation

To find the limit of the function (sqrt(1+2x) - sqrt(1+x)) / (sqrt(9+2x) - sqrt(9-x)) as x approaches a certain value, we can use algebraic manipulation and the concept of conjugate pairs to simplify the expression and evaluate the limit.

First, let's simplify the given function by rationalizing the numerator and the denominator. Rationalizing the numerator involves multiplying the numerator and the denominator by the conjugate of the numerator, and rationalizing the denominator involves multiplying the numerator and the denominator by the conjugate of the denominator.

The conjugate of a binomial expression (a - b) is (a + b).

Rationalizing the Numerator and Denominator

Rationalizing the numerator: sqrt(1+2x) - sqrt(1+x) can be rationalized by multiplying the numerator and the denominator by (sqrt(1+2x) + sqrt(1+x)).

Rationalizing the denominator: sqrt(9+2x) - sqrt(9-x) can be rationalized by multiplying the numerator and the denominator by (sqrt(9+2x) + sqrt(9-x)).

Simplified Function

After rationalizing the numerator and the denominator, the function becomes: (sqrt(1+2x) - sqrt(1+x)) / (sqrt(9+2x) - sqrt(9-x)) = ((sqrt(1+2x) - sqrt(1+x))(sqrt(9+2x) + sqrt(9-x))) / ((sqrt(9+2x) - sqrt(9-x))(sqrt(9+2x) + sqrt(9-x)))

Further Simplification

By multiplying the conjugate pairs, the expression simplifies to: (2x) / (10)

Limit Calculation

Now, as x approaches any value, the limit of the function (sqrt(1+2x) - sqrt(1+x)) / (sqrt(9+2x) - sqrt(9-x)) is: 2x / 10

This means that the limit of the given function as x approaches any value is (x / 5).