Question

Lim x^3+x^2-5x+3/x^3-x^2-x+1 x--->1

Answer 1

Ответ 2................

Answer 2

To find the limit of the given function as x approaches 1, we can directly substitute x = 1 into the function and simplify:

f(x) = (x^3 + x^2 - 5x + 3) / (x^3 - x^2 - x + 1)

Let's evaluate f(1):

f(1) = (1^3 + 1^2 - 5(1) + 3) / (1^3 - 1^2 - 1 + 1) = (1 + 1 - 5 + 3) / (1 - 1 - 1 + 1) = (0) / (0)

We have a 0/0 indeterminate form, which means we need to apply some algebraic manipulation to simplify the expression further and find the limit. To do that, we can factor both the numerator and denominator and cancel out any common factors:

f(x) = [(x - 1)(x^2 + 2x - 3)] / [(x - 1)(x^2 + 1)]

Now, we can cancel out the common factor (x - 1):

f(x) = (x^2 + 2x - 3) / (x^2 + 1)

Now, let's evaluate the limit as x approaches 1 again:

lim(x -> 1) f(x) = lim(x -> 1) [(x^2 + 2x - 3) / (x^2 + 1)]

Since we have removed the indeterminate form, we can directly substitute x = 1:

lim(x -> 1) f(x) = [(1^2 + 2(1) - 3) / (1^2 + 1)] = (1 + 2 - 3) / (1 + 1) = 0 / 2 = 0

So, the limit of the function as x approaches 1 is 0.