Sure! Let's find the limit of the function [tex]\(\frac{3m^2 - 3}{m - 1}\)[/tex] as [tex]\( m \)[/tex] approaches 1.
1. Identify the function and the limit to be evaluated:
We need to evaluate:
[tex]\[
\lim_{m \to 1} \frac{3m^2 - 3}{m - 1}
\][/tex]
2. Assess direct substitution:
Start by substituting [tex]\( m = 1 \)[/tex] directly into the function:
[tex]\[
\frac{3(1)^2 - 3}{1 - 1} = \frac{3 - 3}{0} = \frac{0}{0}
\][/tex]
The result is an indeterminate form, [tex]\( \frac{0}{0} \)[/tex]. Therefore, we need to use algebraic manipulation to simplify the expression.
3. Factor the numerator, if possible:
Factor the numerator [tex]\( 3m^2 - 3 \)[/tex]:
[tex]\[
3m^2 - 3 = 3(m^2 - 1)
\][/tex]
Notice that [tex]\( m^2 - 1 \)[/tex] can be factored further as a difference of squares:
[tex]\[
m^2 - 1 = (m - 1)(m + 1)
\][/tex]
Therefore:
[tex]\[
3m^2 - 3 = 3(m - 1)(m + 1)
\][/tex]
4. Simplify the overall expression:
Substitute the factored form back into the original function:
[tex]\[
\frac{3(m - 1)(m + 1)}{m - 1}
\][/tex]
Now, if [tex]\( m \neq 1 \)[/tex], we can cancel the common factor of [tex]\( m - 1 \)[/tex]:
[tex]\[
\frac{3 \cancel{(m - 1)} (m + 1)}{\cancel{m - 1}} = 3(m + 1)
\][/tex]
5. Evaluate the limit of the simplified expression as [tex]\( m \to 1 \)[/tex]:
Now, as [tex]\( m \)[/tex] approaches 1:
[tex]\[
\lim_{m \to 1} 3(m + 1) = 3(1 + 1) = 3 \cdot 2 = 6
\][/tex]
Therefore, the limit is:
[tex]\[
\lim_{m \to 1} \frac{3m^2 - 3}{m - 1} = 6
\][/tex]
So, the result is 6.
