One Sided Limits of Function Involving Absolute Value and Cosine Term

To determine which assertions hold true, we need to evaluate the left-hand limit ($\lim_{x \to 1^-} f(x)$) and the right-hand limit ($\lim_{x \to 1^+} f(x)$) of the function:

$f(x) = \frac{1-x(1+|1-x|)}{|1-x|} \cos\left(\frac{1}{1-x}\right)$

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1. Evaluation of Left-Hand Limit ($\lim_{x \to 1^-} f(x)$):

When $x \to 1^-$, we have $x < 1$, which implies $1 - x > 0$. Therefore, $|1 - x| = 1 - x$.

Substituting $|1 - x| = 1 - x$ into $f(x)$:

$f(x) = \frac{1 - x(1 + 1 - x)}{1 - x} \cos\left(\frac{1}{1-x}\right)$ $f(x) = \frac{1 - x(2 - x)}{1 - x} \cos\left(\frac{1}{1-x}\right)$ $f(x) = \frac{1 - 2x + x^2}{1 - x} \cos\left(\frac{1}{1-x}\right)$ $f(x) = \frac{(1 - x)^2}{1 - x} \cos\left(\frac{1}{1-x}\right)$ $f(x) = (1 - x) \cos\left(\frac{1}{1-x}\right)$

Now, taking the limit as $x \to 1^-$: Since $-1 \le \cos\left(\frac{1}{1-x}\right) \le 1$ for all $x \neq 1$, the cosine term is bounded.

By the Squeeze Theorem: $-(1 - x) \le (1 - x) \cos\left(\frac{1}{1-x}\right) \le (1 - x)$

Taking the limit as $x \to 1^-$: $\lim_{x \to 1^-} [-(1 - x)] = 0 \quad \text{and} \quad \lim_{x \to 1^-} (1 - x) = 0$

Hence, $\lim_{x \to 1^-} f(x) = 0$

Thus, Option (A) is correct, and Option (B) is incorrect.

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2. Evaluation of Right-Hand Limit ($\lim_{x \to 1^+} f(x)$):

When $x \to 1^+$, we have $x > 1$, which implies $1 - x < 0$. Therefore, $|1 - x| = -(1 - x) = x - 1$.

Substituting $|1 - x| = x - 1$ into $f(x)$:

$f(x) = \frac{1 - x(1 + x - 1)}{x - 1} \cos\left(\frac{1}{1-x}\right)$ $f(x) = \frac{1 - x^2}{x - 1} \cos\left(\frac{1}{1-x}\right)$ $f(x) = \frac{-(x - 1)(x + 1)}{x - 1} \cos\left(\frac{1}{1-x}\right)$ $f(x) = -(x + 1) \cos\left(\frac{1}{1-x}\right)$

Now, taking the limit as $x \to 1^+$:

• As $x \to 1^+$, $-(x + 1) \to -2 \neq 0$.
• As $x \to 1^+$, $\frac{1}{1 - x} \to -\infty$.

The term $\cos\left(\frac{1}{1-x}\right)$ oscillates infinitely between $-1$ and $1$ as $x \to 1^+$. Consequently, $f(x)$ oscillates infinitely between $-2$ and $2$.

Therefore, $\lim_{x \to 1^+} f(x)$ does not exist.

Thus, Option (D) is correct, and Option (C) is incorrect.

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Conclusion:

The correct statements are A and D.