Limit of (1 − cos^N x) over x²
Evaluate the limit, for a fixed real constant $N\neq0$: $$ L = \lim_{x \to 0} \frac{1-\cos^N x}{x^2}. $$
Explanation of Method
Direct substitution gives $\frac{1-1}{0}=\frac00$. Writing $\cos x=1-\frac{x^2}2+O(x^4)$ and applying the binomial approximation $(1-u)^N\approx1-Nu$ for small $u$ (valid for any real exponent $N$, not just integers) shows that raising $\cos x$ to the power $N$ simply multiplies the leading correction term by $N$. This immediately generalizes the familiar case $N=1$ (where $1-\cos x\sim\frac{x^2}2$) to any power.
- Expand $\cos x$ using its Taylor series.
- Apply the binomial approximation to $\cos^Nx=(1-\tfrac{x^2}2+\cdots)^N$.
- Substitute into the numerator and divide by $x^2$.
Step 1 · Expand $\cos x$
The Taylor series for $\cos x$ near $0$ is:
$$ \cos x = 1-\frac{x^2}{2}+O(x^4). $$
Step 2 · Apply the binomial approximation to $\cos^Nx$
Using the binomial approximation $(1-u)^N\approx1-Nu+\dfrac{N(N-1)}2u^2+O(u^3)$ with $u=\dfrac{x^2}2$:
$$ \cos^Nx = \left(1-\frac{x^2}2\right)^N \approx 1-\frac{Nx^2}2+\frac{N(N-1)x^4}8+O(x^6). $$
Step 3 · Substitute into the numerator and divide by $x^2$
Subtracting from $1$:
$$ 1-\cos^Nx \approx \frac{Nx^2}2-\frac{N(N-1)x^4}8+O(x^6). $$
Dividing by $x^2$:
$$ \frac{1-\cos^Nx}{x^2} \approx \frac{N}{2}-\frac{N(N-1)x^2}8+O(x^4). $$
As $x\to0$, the correction terms vanish, leaving:
$$ L = \lim_{x\to0}\frac{1-\cos^Nx}{x^2} = \frac{N}{2}. $$
Summary Table
| Quantity | Approximation for small $x$ |
|---|---|
| $\cos x$ | $1-\dfrac{x^2}2+O(x^4)$ |
| $\cos^Nx$ | $1-\dfrac{Nx^2}2+O(x^4)$ |
| $\dfrac{1-\cos^Nx}{x^2}$ | $\dfrac N2+O(x^2) \ \to\ \dfrac N2$ |
Additional Notes
- Setting $N=1$ recovers the familiar rate $1-\cos x\sim\frac{x^2}2$, i.e. $\lim_{x\to0}\frac{1-\cos x}{x^2}=\frac12$, seen elsewhere on this site — this problem shows that result is just the special case of a much more general one.
- The binomial approximation $(1-u)^N\approx1-Nu$ holds for any real exponent $N$ (not just positive integers), which is why this result is stated for a general real constant $N\neq0$ rather than just a positive integer power.
- For negative $N$ (for instance $N=-1$, giving $\frac1{\cos x}=\sec x$), the same formula applies directly: $\lim_{x\to0}\frac{1-\sec x}{x^2}=\frac{-1}2=-\frac12$, matching what a direct expansion of $\sec x = 1+\frac{x^2}2+O(x^4)$ would also confirm.
Result
The value of the limit is: