Limit of sinh x − x as x approaches 0
Evaluate the limit of $Y(x) = \sinh(x) - x$ as $x \to 0$.
Explanation of Method
Since $\sinh$ and the identity function $x$ are both continuous everywhere, the limit of their difference as $x\to0$ follows immediately by direct substitution. As with the analogous cosine problem, the more informative question is how quickly $Y(x)$ vanishes; expanding $\sinh x$ with its Taylor series shows that the linear terms cancel exactly, leaving a leading behavior of $\frac{x^3}{6}$.
- Evaluate the limit directly using continuity of $\sinh$ and of the identity function.
- Expand $\sinh x$ with its Taylor series to find the precise rate at which $Y(x)\to0$.
Step 1 · Direct evaluation using continuity
Recall $\sinh(x) = \dfrac{e^x-e^{-x}}{2}$, which is continuous everywhere (being built from continuous exponentials), with $\sinh(0)=\dfrac{1-1}{2}=0$. So:
$$ \lim_{x\to0}\sinh(x) = \sinh(0) = 0. $$
Therefore:
$$ \lim_{x\to0} Y(x) = \lim_{x\to0}\left(\sinh x - x\right) = 0 - 0 = 0. $$
Step 2 · The rate of vanishing (Taylor expansion)
The Taylor series for $\sinh x$ about $x=0$ is:
$$ \sinh x = x + \frac{x^3}{6} + \frac{x^5}{120} + \cdots. $$
Substituting into $Y(x)$:
$$ Y(x) = \sinh x - x = \left(x+\frac{x^3}{6}+\frac{x^5}{120}+\cdots\right) - x = \frac{x^3}{6}+\frac{x^5}{120}+\cdots. $$
The linear term $x$ cancels exactly, leaving the leading behavior:
$$ Y(x) \sim \frac{x^3}{6} \qquad \text{for } x \text{ near } 0, $$
confirming $\lim_{x\to0} Y(x) = \lim_{x\to0}\dfrac{x^3}{6}=0$, consistent with Step 1.
Summary Table
| Approach | Key fact used | Result |
|---|---|---|
| Direct substitution | Continuity of $\sinh$ at $0$ | $0$ |
| Taylor expansion | $\sinh x - x \sim \dfrac{x^3}{6}$ | $0$, with cubic rate |
Additional Notes
- The exact cancellation of the linear term $x$ in the expansion of $\sinh x - x$ mirrors the analogous cancellation of the constant term in $1-\cos x$: in each case, subtracting off the lowest-order term of the Taylor series leaves a "remainder" function whose leading behavior is governed by the next surviving power.
- Because the leading term here is $\frac{x^3}{6}$ — an odd power — $Y(x)=\sinh x - x$ is an odd function, changing sign with $x$; this matches the fact that $\sinh$ itself is odd while $x$ is trivially odd, so their difference is odd as well.
- This cubic rate of vanishing, $\sinh x - x \sim \frac{x^3}{6}$, is a standard building block used when simplifying more complex limits that contain $\sinh x - x$ as a factor, analogous to how $1-\cos x\sim\frac{x^2}{2}$ or $x-\sin x\sim\frac{x^3}{6}$ are used elsewhere.
Result
The value of the limit is: