Certainly! Let’s evaluate the limit:
[tex]\[ \lim_{{x \to 1}} \frac{x^4 + x^3 + x^2 + x - 4}{x - 1} \][/tex]
### Step 1: Recognize Indeterminate Form
First, plug [tex]\( x = 1 \)[/tex] into the expression to see what form we get:
[tex]\[ \frac{1^4 + 1^3 + 1^2 + 1 - 4}{1 - 1} = \frac{1 + 1 + 1 + 1 - 4}{0} = \frac{0}{0} \][/tex]
This is an indeterminate form, so we need to simplify or transform the expression to evaluate the limit.
### Step 2: Factor the Numerator
Since direct substitution gives an indeterminate form, we need to factor the numerator and see if there's a way to cancel out the [tex]\( x - 1 \)[/tex] in the denominator.
Consider the expression [tex]\( x^4 + x^3 + x^2 + x - 4 \)[/tex]. We want to factor it such that we can divide out the [tex]\( x - 1 \)[/tex].
### Step 3: Polynomial Division (optional method)
Alternatively, since this division isn't straightforward to factor by inspection, we use polynomial division or synthetic division to rewrite the numerator as a product involving [tex]\( x - 1 \)[/tex].
But for conciseness, we might skip the long division steps here given that it leads us to the factored form. When [tex]\( x = 1 \)[/tex] is a root, our polynomial [tex]\( x^4 + x^3 + x^2 + x - 4 \)[/tex] can be expressed as:
[tex]\[ (x - 1)Q(x) \][/tex]
where [tex]\( Q(x) \)[/tex] is a polynomial of degree 3.
### Step 4: Direct Simplification and the Limit
Once we factor and cancel the common term [tex]\( x - 1 \)[/tex] in the numerator and denominator, we are left with:
[tex]\[ \lim_{{x \to 1}} Q(x) \][/tex]
If we perform this factorization (which can be checked using division or algebraic manipulation principles), we would find:
[tex]\[ x^4 + x^3 + x^2 + x - 4 = (x - 1)(x^3 + 2x^2 + 3x + 4) \][/tex]
### Step 5: Substitute and Simplify
Thus, our limit becomes:
[tex]\[ \lim_{{x \to 1}} \frac{(x - 1)(x^3 + 2x^2 + 3x + 4)}{x - 1} = \lim_{{x \to 1}} (x^3 + 2x^2 + 3x + 4) \][/tex]
As [tex]\( x \to 1 \)[/tex], substitute [tex]\( x = 1 \)[/tex] into [tex]\( x^3 + 2x^2 + 3x + 4 \)[/tex]:
[tex]\[ 1^3 + 2(1^2) + 3(1) + 4 = 1 + 2 + 3 + 4 = 10 \][/tex]
Thus, the limit is:
[tex]\[ \boxed{10} \][/tex]
