Limit of (1−x) tan(πx/2) as x approaches 1
Evaluate the limit: $$ L = \lim_{x \to 1} (1-x)\tan\left(\frac{\pi x}{2}\right). $$
Explanation of Method
As $x \to 1$, the factor $1-x \to 0$ while $\tan\left(\frac{\pi x}{2}\right) \to \tan\left(\frac{\pi}{2}\right)$, which blows up to infinity, giving the indeterminate form $0\cdot\infty$. Substituting $y=1-x$ shifts the limit point to $y=0$ and turns the tangent argument into $\frac{\pi}{2}-\frac{\pi y}{2}$, which can be rewritten as a cotangent using the standard co-function identity. Cotangent near $0$ behaves like $\frac{1}{z}$, which combines cleanly with the factor of $y$ out front.
- Substitute $y = 1-x$, so that $y \to 0$ as $x \to 1$.
- Rewrite the tangent using the co-function identity $\tan\left(\frac{\pi}{2}-z\right) = \cot(z)$.
- Approximate $\cot(z) \approx \frac{1}{z}$ for small $z$.
- Simplify the resulting expression and take the limit.
Step 1 · Substitute $y=1-x$
Let $y = 1-x$, so $x = 1-y$, and as $x \to 1$, $y \to 0$. The limit becomes:
$$ L = \lim_{y \to 0} y\cdot \tan\left(\frac{\pi(1-y)}{2}\right) = \lim_{y \to 0} y\cdot\tan\left(\frac{\pi}{2}-\frac{\pi y}{2}\right). $$
Step 2 · Rewrite the tangent as a cotangent
Using the co-function identity $\tan\left(\frac{\pi}{2}-z\right) = \cot(z)$ with $z = \frac{\pi y}{2}$:
$$ \tan\left(\frac{\pi}{2}-\frac{\pi y}{2}\right) = \cot\left(\frac{\pi y}{2}\right). $$
So:
$$ L = \lim_{y \to 0} y\cdot\cot\left(\frac{\pi y}{2}\right). $$
Step 3 · Use the small-argument behavior of cotangent
Since $\cot(z) = \dfrac{\cos z}{\sin z}$ and $\sin z \approx z$ for small $z$ while $\cos z \approx 1$, we have $\cot(z) \approx \dfrac{1}{z}$ as $z \to 0$. With $z = \frac{\pi y}{2} \to 0$ as $y \to 0$:
$$ \cot\left(\frac{\pi y}{2}\right) \approx \frac{1}{\frac{\pi y}{2}} = \frac{2}{\pi y}. $$
Step 4 · Combine and take the limit
Substituting this approximation:
$$ L = \lim_{y \to 0} y \cdot \frac{2}{\pi y} = \lim_{y \to 0} \frac{2}{\pi} = \frac{2}{\pi}. $$
Summary Table
| Quantity | After substitution $y=1-x$ | Behavior as $y \to 0$ |
|---|---|---|
| $1-x$ | $y$ | $\to 0$ |
| $\tan\left(\frac{\pi x}{2}\right)$ | $\cot\left(\frac{\pi y}{2}\right)$ | $\approx \dfrac{2}{\pi y} \to \infty$ |
| Product | $y\cot\left(\frac{\pi y}{2}\right)$ | $\to \dfrac{2}{\pi}$ |
Additional Notes
- The co-function identity $\tan\left(\frac{\pi}{2}-z\right)=\cot(z)$ is essential here: it converts a tangent blowing up near $\frac{\pi}{2}$ into a cotangent blowing up near $0$, where the standard small-angle approximation $\cot z \approx \frac{1}{z}$ applies directly.
- Equivalently, one can write $y\cot\left(\frac{\pi y}{2}\right) = \dfrac{2}{\pi}\cdot\dfrac{\frac{\pi y}{2}}{\sin\left(\frac{\pi y}{2}\right)}\cdot\cos\left(\frac{\pi y}{2}\right) \to \dfrac{2}{\pi}\cdot 1 \cdot 1 = \dfrac{2}{\pi}$, using the standard limit $\lim_{z\to0}\frac{z}{\sin z}=1$ — a fully rigorous alternative to the small-angle approximation used above.
- This type of limit — a linear factor vanishing at the same rate that a trigonometric function blows up — always resolves to a finite constant, determined by the coefficient in the trig function's argument.
Result
The value of the limit is: