Question

Let \(f(x)=\frac{1-x(1+|1-x|)}{|1-x|} \cos \left(\frac{1}{1-x}\right)\) for \(x \neq 1\). Then [A] \(\lim _{x \rightarrow 1^{-}} f(x)=0\) [B] \(\lim _{x \rightarrow 1^{-}} f(x)\) does not exist [C] \(\lim _{x \rightarrow 1^{+}} f(x)=0\) [D] \(\lim _{x \rightarrow 1^{+}} f(x)\) does not exist

Short answer

[A] and [C] are true; both left and right limits as x approaches 1 for f(x) are 0.

Step-by-step solution

Simplify the Function for x Approaching 1 from the Left

First, observe that as x approaches 1 from the left (i.e., values of x are slightly less than 1), the value of |1-x| is simply 1-x since it is positive. This will help in simplifying the expression for f(x) when considering limit from the left.

Calculate the Left-Hand Limit

Now, plug in values of x that are slightly less than 1 into the simplified expression of f(x). Noticing that \(\cos\left(\frac{1}{1-x}\right)\) is bounded between -1 and 1, regardless of the value of x, it will not affect the limit as x approaches 1 since the multiplier \(\frac{1-x(1+(1-x))}{1-x}\) approaches 0. Therefore, the left-hand limit of f(x) as x approaches 1 is 0.

Simplify the Function for x Approaching 1 from the Right

For x slightly greater than 1, |1-x| becomes x-1 because the expression inside the absolute value is negative. Again, simplify the expression for f(x) when considering the limit from the right.

Calculate the Right-Hand Limit

For values of x just larger than 1, directly substitute these values into the simplified f(x). Similar to the left-hand limit, the \(\cos\left(\frac{1}{1-x}\right)\) term does not impact the limit due to its bounded nature. The multiplier \(\frac{1-x(1-(x-1))}{x-1}\) once again approaches 0 as x approaches 1 from the right, so the right-hand limit of f(x) is also 0.

Conclude the Limits

Since both the left-hand and right-hand limits of f(x) as x approaches 1 are 0, statements [A] and [C] are true.

Key concepts

Left-Hand Limit

In the realm of calculus, understanding the concept of a left-hand limit is fundamental. It refers to the value that a function approaches as the input approaches a certain point from the left side on a number line. To visualize it, imagine walking along a graph towards a point just to the left of a specified number. Formally, the left-hand limit of a function at a certain point, say 'a', is written as \( \[\lim _{x \rightarrow a^{-}} f(x)\] \). It's crucial to pay attention to the behavior of the function as 'x' gets infinitesimally close to 'a' but remains less than 'a'.

When solving such limits, it's important to consider any simplifications that can ease the calculation. For instance, dealing with absolute values can often help break down the expression into more manageable pieces. In the exercise provided, identifying that \( |1-x| = 1-x \) when approaching 1 from the left aids in simplifying the given function before calculating the limit. Thus, as 'x' nudges closer to 1 from the left, the function's expression becomes less complex and the ultimate value of the limit can be determined.

Right-Hand Limit

Complementing the left-hand limit is the right-hand limit, which examines what happens to the function's outputs as the input comes close to a given value from the right. This means we investigate the function's behavior as 'x' approaches a point, say 'b', from values greater than 'b'. The notation \( \[\lim _{x \rightarrow b^{+}} f(x)\] \) represents this limit.

To evaluate a right-hand limit, the method usually involves substituting values into the function that are slightly greater than the point of interest and then analyzing the result. As in the example provided, when considering \( |1-x| \) as 'x' approaches 1 from the right, it simplifies to \( x-1 \) because we are dealing with positive values. Simplifying the function in this way is crucial for an accurate determination of the right-hand limit, which can often mirror the process for finding the left-hand limit.

Absolute Value in Limits

The absolute value in limits can significantly influence the function's behavior near the point of interest and, therefore, alter the limit's value. The absolute value operation, denoted by | |, essentially provides the distance of a number from zero on the number line, always as a non-negative value. In the context of limits, when variables inside the absolute value approach the point from different directions, the outcomes may differ.

For example, \( |1-x| \) behaves differently depending on whether 'x' is approaching 1 from the left or the right. This nuance affects how we simplify the function prior to calculating the limit. The breakdown of absolute value into piecewise functions depending on the input value is particularly useful in tackling limits where absolute values are present, as demonstrated in the exercise.

Boundedness of Trigonometric Functions

Trigonometric functions, like sine and cosine, are inherently bounded, which means that their outputs are restricted to a certain range. Specifically, the outputs for these functions lie between -1 and 1, inclusive. This property is extremely useful when evaluating limits, especially when the trigonometric function is multiplied by another term that tends to zero.

No matter how complex the expressions within these trigonometric functions get—such as \( \cos(\frac{1}{1-x}) \) in our exercise—their boundedness assures us that their contribution to the product in the limit will not push the result toward infinity or cause it to oscillate uncontrollably. Thus, if the other term in the product approaches zero, the entire expression's limit can be anticipated to be zero, just as we observed in the sample problem for both the left-hand and right-hand limits.