Question

Найти предел функцииРешение пожалуйста письменно

Answer 1

Смотри решение во вложении

Answer 2

Finding the Limit of a Function

To find the limit of a function, we can use various methods such as direct substitution, factoring, rationalizing the numerator, or using L'Hôpital's rule. Let's consider an example to illustrate the process.

Suppose we want to find the limit of the function f(x) = (x^2 - 4) / (x - 2) as x approaches 2.

Solution:

We can start by applying direct substitution to evaluate the limit. When we substitute x = 2 into the function, we get:

f(2) = (2^2 - 4) / (2 - 2) = (4 - 4) / 0

This results in an indeterminate form of 0/0. To resolve this, we can factor the numerator and simplify the expression.

Factoring the numerator gives us:

f(x) = ((x - 2)(x + 2)) / (x - 2)

Now, we can cancel out the common factor of (x - 2):

f(x) = x + 2

Now, we can use direct substitution again to find the limit:

lim(x->2) (x + 2) = 2 + 2 = 4

So, the limit of the function f(x) = (x^2 - 4) / (x - 2) as x approaches 2 is 4.

This example demonstrates the process of finding the limit of a function using factoring and direct substitution.