Sure, let's go through the solution step-by-step to find the limit of the function as [tex]\( x \)[/tex] approaches 0:
We are given the function:
[tex]\[ f(x) = \frac{x^2}{x^2 + 1} \][/tex]
We need to find the limit of this function as [tex]\( x \)[/tex] approaches 0:
[tex]\[ \lim_{x \to 0} \frac{x^2}{x^2 + 1} \][/tex]
To do this, let's analyze the behavior of the numerator and the denominator separately as [tex]\( x \)[/tex] approaches 0.
1. Numerator Analysis:
The numerator of our function is [tex]\( x^2 \)[/tex]. As [tex]\( x \)[/tex] approaches 0, [tex]\( x^2 \)[/tex] (which is [tex]\( x \)[/tex] squared) also approaches 0. So, the numerator approaches 0.
2. Denominator Analysis:
The denominator is [tex]\( x^2 + 1 \)[/tex]. As [tex]\( x \)[/tex] approaches 0, the term [tex]\( x^2 \)[/tex] approaches 0, and hence the denominator approaches [tex]\( 0 + 1 = 1 \)[/tex].
Now, combining these two results:
- As [tex]\( x \)[/tex] approaches 0, the numerator [tex]\( x^2 \)[/tex] approaches 0.
- As [tex]\( x \)[/tex] approaches 0, the denominator [tex]\( x^2 + 1 \)[/tex] approaches 1.
Thus, our limit expression becomes:
[tex]\[
\lim_{x \to 0} \frac{x^2}{x^2 + 1} = \frac{0}{1} = 0
\][/tex]
Therefore, the limit is:
[tex]\[
\boxed{0}
\][/tex]
