Evaluate the limit of 3 as x approaches infinity.

Step-by-step solution

1. 1

   Identify the function. The expression inside the limit contains no $x$ at all, so it is the constant function $f(x)=3$. Its graph is a horizontal line at height $3$.

2. 2

   Apply the constant rule for limits. For any constant $c$ and any approach — a finite point or infinity —

   $\lim_{x\to a}c=c$

   because the function's output never changes as the input moves.

3. 3

   Conclude.

   $\lim_{x\to\infty}3=3$

4. 4

   Check it against the definition. A limit $L$ at infinity requires that for every $\varepsilon>0$ there is an $M$ with $|f(x)-L|<\varepsilon$ whenever $x>M$. Here $|3-3|=0<\varepsilon$ for every $x$, so any $M$ works — the condition is satisfied trivially.

5. 5

   Note the common confusion. The answer is not $\infty$: it is the input that grows without bound, not the output. Nor is it undefined — constants are among the best-behaved functions there are. Compare $\lim_{x\to\infty}\tfrac{3x+1}{x}=3$, which also equals $3$ but for a genuinely different reason: there the function moves toward $3$, whereas here it sits at $3$ throughout.