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How do I draw the graph of a function when the limit is given? For example, how can I draw the graph of the following function:

$$f(x)=\lim_{n\to∞ }\frac{x^{2n+1}}{x^{2n}+1}$$

if the value of x is less than 1 the f(x) should decrease. And it will increase while it is more than 1. But I am not sure whether it will be a linear graph or a curve.

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up vote 2 down vote accepted

$\lim_{n\to∞ }\frac{x^{2n+1}}{x^{2n}+1}$ is clearly $0$ for $x \in [0,1)$ (as both $x^{2n+1}$ and $x^{2n}$ tend to zero as $n \to \infty$), then it is equal to $\frac{1}{2}$ for $x = 1$, and for every $x > 1$, since $\lim_{n\to∞ }\frac{x^{2n+1}}{x^{2n}+1} =\lim_{n\to∞ } \frac{x}{1 + \frac{1}{x^{2n}}} = x$, it is equal to $x$.

Hence, $f(x) = 0$ for $x \in [0,1)$, $f(x) = \frac{1}{2}$ for $x=1$, and $f(x) = x$ for every $x > 1$.

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