Let's solve the limit [tex]\(\lim _{x \rightarrow 1}\left(\frac{1}{1-x}-\frac{3}{1-x^3}\right)\)[/tex] step by step.
Firstly, examine the function given in the limit:
[tex]\[ f(x) = \frac{1}{1-x} - \frac{3}{1-x^3} \][/tex]
We need to find the limit of [tex]\(f(x)\)[/tex] as [tex]\(x\)[/tex] approaches 1.
Step 1: Simplify the expression
Notice that the second term in the expression has a denominator [tex]\(1 - x^3\)[/tex]. We can factorize [tex]\(1 - x^3\)[/tex] as follows:
[tex]\[ 1 - x^3 = (1-x)(1 + x + x^2) \][/tex]
This means we can rewrite the second term:
[tex]\[ \frac{3}{1-x^3} = \frac{3}{(1-x)(1 + x + x^2)} \][/tex]
Now our function becomes:
[tex]\[ f(x) = \frac{1}{1-x} - \frac{3}{(1-x)(1 + x + x^2)} \][/tex]
Step 2: Combine the fractions
Both terms now have a common factor in the denominator, which is [tex]\(1 - x\)[/tex]. We can combine them:
[tex]\[ f(x) = \frac{1(1 + x + x^2) - 3}{(1-x)(1+x+x^2)} \][/tex]
Step 3: Simplify the numerator
Simplify the numerator [tex]\(1(1 + x + x^2) - 3\)[/tex]:
[tex]\[ 1 + x + x^2 - 3 = x + x^2 - 2 \][/tex]
Our function now looks like this:
[tex]\[ f(x) = \frac{x + x^2 - 2}{(1-x)(1+x+x^2)} \][/tex]
Step 4: Factorize the numerator [tex]\(x + x^2 - 2\)[/tex]
To factorize [tex]\(x + x^2 - 2\)[/tex], observe that:
[tex]\[ x^2 + x - 2 = (x-1)(x+2) \][/tex]
So the function becomes:
[tex]\[ f(x) = \frac{(x-1)(x+2)}{(1-x)(1+x+x^2)} \][/tex]
Step 5: Simplify the function further
Notice that [tex]\((x-1)\)[/tex] and [tex]\((1-x)\)[/tex] are opposites of each other. We know that:
[tex]\[ 1-x = -(x-1) \][/tex]
So we can rewrite our function:
[tex]\[ f(x) = \frac{(x-1)(x+2)}{-(x-1)(1+x+x^2)} \][/tex]
The [tex]\((x-1)\)[/tex] terms cancel out:
[tex]\[ f(x) = \frac{x+2}{-(1+x+x^2)} \][/tex]
Which simplifies to:
[tex]\[ f(x) = -\frac{x+2}{1 + x + x^2} \][/tex]
Step 6: Evaluate the limit as [tex]\(x \rightarrow 1\)[/tex]
Now, substitute [tex]\(x = 1\)[/tex] in the simplified function:
[tex]\[ \lim_{x \to 1} f(x) = -\frac{1+2}{1 + 1 + 1^2} = -\frac{3}{3} = -1 \][/tex]
Thus, the limit is:
[tex]\[ \lim _{x \rightarrow 1}\left(\frac{1}{1-x}-\frac{3}{1-x^3}\right) = -1 \][/tex]
So, the final result is:
[tex]\[ \boxed{-1} \][/tex]
