Animated Solution for Mathematics - Limits, Continuity and Differentiability: ${\lim }_{x\to \frac{1}{\sqrt{2}}}\frac{\sin ({\cos }^{-1}x)-x}{1-\tan ({\cos }^{-1}x)}$ is equal to :

Untangling the Inverse

When you first look at a limit like

it is natural to feel a surge of intimidation. We see inverse trigonometric functions nested inside standard trigonometric functions, and our instinct might be to panic.

But in the world of JEE Advanced, this is not a wall; it is a puzzle. The key is to stop seeing these as abstract operators and start seeing them as geometric relationships.

The Substitution Strategy

Let us simplify our lives. We define $\theta ={\cos }^{-1}x$. This is the most powerful move in our arsenal.

By definition, this implies $\cos \theta =x$. Imagine a right-angled triangle where the base angle is $\theta$. If $\cos \theta =\frac{\text{base}}{\text{hypotenuse}}=x$, we can set the base to $x$ and the hypotenuse to $1$.

By the Pythagorean theorem, the perpendicular side becomes $\sqrt{1-x^2}$. Now, look at the expression $\sin ({\cos }^{-1}x)$. This is simply $\sin \theta$. From our triangle, $\sin \theta =\sqrt{1-x^2}$. We have successfully stripped away the inverse function!

The Boundary Shift

We must be precise. When we change our variable from $x$ to $\theta$, we must change the limit boundary.

Our original limit states $x\to \frac{1}{\sqrt{2}}$. Since $\cos \theta =x$, we ask: at what angle $\theta$ does $\cos \theta$ equal $\frac{1}{\sqrt{2}}$?

The answer is $\frac{\pi }{4}$. Thus, as $x\to \frac{1}{\sqrt{2}}$, our new boundary is $\theta \to \frac{\pi }{4}$.

The Algebraic Dance

Now, let us rewrite our limit entirely in terms of $\theta$:

This looks much cleaner, but if we substitute $\theta =\frac{\pi }{4}$ immediately, we still face a $0/0$ indeterminate form. We need to go deeper.

Let us expand $\tan \theta$ as $\frac{\sin \theta }{\cos \theta }$. The denominator becomes $1-\frac{\sin \theta }{\cos \theta }$, which simplifies to $\frac{\cos \theta -\sin \theta }{\cos \theta }$.

Now, substitute this back into our limit:

When we divide by a fraction, we multiply by its reciprocal. This brings the $\cos \theta$ to the numerator:

Here is the moment of truth. Notice that $(\cos \theta -\sin \theta )$ is just the negative of $(\sin \theta -\cos \theta )$.

We can rewrite the denominator as $-(\sin \theta -\cos \theta )$. The common factor $(\sin \theta -\cos \theta )$ cancels out perfectly, leaving us with:

Finally, we evaluate the limit by substituting $\theta =\frac{\pi }{4}$. Since $\cos (\frac{\pi }{4})=\frac{1}{\sqrt{2}}$, our final result is $-\frac{1}{\sqrt{2}}$.

We have navigated the complexity, simplified the geometry, and arrived at the solution with elegance. Keep this mindset—break the problem down, and the math will always reveal its path.