\( \operatorname{im}_{x \rightarrow 0}\left(x^{2}+1\right) \frac{\tan x}{x} \quad \) use properties of limits

To find the limit as \( x \) approaches \( 0 \) for the expression \( \left(x^{2}+1\right) \frac{\tan x}{x} \), we can analyze each component. First, as \( x \) approaches \( 0 \), \( x^{2} + 1 \) approaches \( 1 \). Next, \( \frac{\tan x}{x} \) is a well-known limit that approaches \( 1 \) as \( x \) approaches \( 0 \). Hence, we can combine these limits: \[ \lim_{x \rightarrow 0} \left(x^{2}+1\right) \frac{\tan x}{x} = (1) \cdot (1) = 1. \] Therefore, the limit evaluates to \( 1 \). Now, don't you just love how math has these interconnected parts? It’s like a puzzle where each piece fits perfectly! Remember, the beauty of limits lies in their simplicity, especially with functions behaving nicely near the points of interest.