Question

Помогите пожалуйста решить границу функций. lim (sqrt(9x^2+2)-x)/(4x+11) x⇒+∞ Буду очень благодарен.

Answer 1

Чтобы найти предел при x -> беск, нужно
Разделить все члены на старшую степень x
у нас это x в 1
$( \sqrt{9*x^2/x^2+2/x^2} -x/x)/(4x/x+11/x)$
при вносе под корень x^2
получается $\frac{ \sqrt{9}-1 }{4}=1/2$

Answer 2

Function Limit Calculation

To solve the limit of the given function as x approaches infinity, we can use algebraic manipulation and the properties of limits. Let's break down the steps to solve the limit.

1. Identify the Function: The given function is: \[ \lim_{{x \to \infty}} \frac{{\sqrt{9x^2+2}-x}}{{4x+11}} \]

2. Rationalizing the Numerator: We can rationalize the numerator by multiplying both the numerator and the denominator by the conjugate of the numerator, which is \(\sqrt{9x^2+2}+x\).

3. Simplify the Expression: After rationalizing the numerator, we simplify the expression to make it easier to evaluate the limit.

4. Apply the Limit: Finally, we can apply the limit as x approaches infinity to the simplified expression to find the value of the limit.

Let's proceed with the rationalization and simplification of the given function.

Rationalizing the Numerator

To rationalize the numerator, we multiply both the numerator and the denominator by the conjugate of the numerator, which is \(\sqrt{9x^2+2}+x\).

The rationalized form of the function becomes: \[ \lim_{{x \to \infty}} \frac{{\sqrt{9x^2+2}-x}}{{4x+11}} \times \frac{{\sqrt{9x^2+2}+x}}{{\sqrt{9x^2+2}+x}} \]

Simplifying the Expression

After rationalizing the numerator, we simplify the expression to make it easier to evaluate the limit.

The simplified form of the function is: \[ \lim_{{x \to \infty}} \frac{{(9x^2+2)-x^2}}{{(4x+11)(\sqrt{9x^2+2}+x)}} \]

Applying the Limit

Now, we can apply the limit as x approaches infinity to the simplified expression to find the value of the limit.

Let's calculate the limit and find the value of the given function as x approaches infinity.