Key Concepts and Formulas

• Inverse Trigonometric Functions: Understanding the definition and range of inverse trigonometric functions, particularly $\cos^{-1}x$.
• Limit Evaluation Techniques: Familiarity with substitution, algebraic manipulation, and trigonometric identities for evaluating limits.
• Trigonometric Identities: Knowledge of basic trigonometric identities, such as $\tan \theta = \frac{\sin \theta}{\cos \theta}$ and $\sin(\frac{\pi}{4}) = \cos(\frac{\pi}{4}) = \frac{1}{\sqrt{2}}$.

Step-by-Step Solution

Step 1: Analyze the Limit Expression and Identify Indeterminacy The given limit is: $L = \mathop {\lim }\limits_{x \to {1 \over {\sqrt 2 }}} {{\sin ({{\cos }^{ - 1}}x) - x} \over {1 - \tan ({{\cos }^{ - 1}}x)}}$ As $x \to \frac{1}{\sqrt{2}}$, we have $\cos^{-1}x \to \cos^{-1}(\frac{1}{\sqrt{2}}) = \frac{\pi}{4}$. Substituting these values into the expression: Numerator: $\sin(\frac{\pi}{4}) - \frac{1}{\sqrt{2}} = \frac{1}{\sqrt{2}} - \frac{1}{\sqrt{2}} = 0$. Denominator: $1 - \tan(\frac{\pi}{4}) = 1 - 1 = 0$. Since we have the indeterminate form $\frac{0}{0}$, we need to use algebraic manipulation or other limit techniques.

Step 2: Introduce a Substitution to Simplify the Expression To simplify the presence of the inverse trigonometric function, let $t = \cos^{-1}x$. This implies $x = \cos t$. We also need to determine the limit of $t$ as $x$ approaches $\frac{1}{\sqrt{2}}$. As $x \to \frac{1}{\sqrt{2}}$, $t = \cos^{-1}x \to \cos^{-1}(\frac{1}{\sqrt{2}})$. The principal value of $\cos^{-1}(\frac{1}{\sqrt{2}})$ is $\frac{\pi}{4}$. So, as $x \to \frac{1}{\sqrt{2}}$, $t \to \frac{\pi}{4}$.

Step 3: Rewrite the Limit in Terms of the New Variable 't' Substitute $x = \cos t$ and $\cos^{-1}x = t$ into the original limit expression: $L = \mathop {\lim }\limits_{t \to {\pi \over 4}} {{\sin t - \cos t} \over {1 - \tan t}}$

Step 4: Apply Trigonometric Identity for Tangent Use the identity $\tan t = \frac{\sin t}{\cos t}$ to rewrite the denominator: $L = \mathop {\lim }\limits_{t \to {\pi \over 4}} {{\sin t - \cos t} \over {1 - {{\sin t} \over {\cos t}}}}$

Step 5: Perform Algebraic Simplification in the Denominator Combine the terms in the denominator by finding a common denominator: $L = \mathop {\lim }\limits_{t \to {\pi \over 4}} {{\sin t - \cos t} \over {{{\cos t - \sin t} \over {\cos t}}}}$

Step 6: Simplify the Complex Fraction Multiply the numerator by the reciprocal of the denominator: $L = \mathop {\lim }\limits_{t \to {\pi \over 4}} {(\sin t - \cos t) \cdot (\cos t) \over {\cos t - \sin t}}$

Step 7: Factor out -1 from the Denominator Notice that the term $(\cos t - \sin t)$ in the denominator is the negative of the term $(\sin t - \cos t)$ in the numerator. We can factor out $-1$ from the denominator: $L = \mathop {\lim }\limits_{t \to {\pi \over 4}} {{(\sin t - \cos t) \cdot (\cos t)} \over {-(\sin t - \cos t)}}$

Step 8: Cancel Common Factors and Evaluate the Limit For $t$ approaching $\frac{\pi}{4}$ but not equal to $\frac{\pi}{4}$, $(\sin t - \cos t) \neq 0$, so we can cancel this term from the numerator and the denominator: $L = \mathop {\lim }\limits_{t \to {\pi \over 4}} { - \cos t}$ Now, substitute the limit value $t = \frac{\pi}{4}$ into the simplified expression: $L = - \cos\left({\pi \over 4}\right)$ $L = - {1 \over {\sqrt 2 }}$

Common Mistakes & Tips

• Incorrectly evaluating $\cos^{-1}(\frac{1}{\sqrt{2}})$: Ensure you use the principal value, which is $\frac{\pi}{4}$ for $\cos^{-1}x$.
• Algebraic errors during simplification: Pay close attention to signs when manipulating fractions and factoring. The step where $(\cos t - \sin t)$ is rewritten as $-(\sin t - \cos t)$ is crucial.
• Forgetting to substitute back or evaluate: After simplifying, always remember to substitute the limit value to find the numerical answer.

Summary

The problem involves evaluating a limit of a trigonometric expression. By recognizing the indeterminate form $\frac{0}{0}$, we employed a substitution $t = \cos^{-1}x$ to transform the limit into a function of $t$. We then used trigonometric identities and algebraic manipulation to simplify the expression. Specifically, rewriting $\tan t$ and factoring out $-1$ from the denominator allowed for cancellation of a common term, leading to a straightforward evaluation of the limit by substituting $t = \frac{\pi}{4}$. The final result is $-\frac{1}{\sqrt{2}}$.

The final answer is $\boxed{-{1 \over {\sqrt 2 }}}$.