Limit as x approaches +∞ of x sin (1/x)
To find the limit as x approaches positive infinity of x sin(1/x), we can use the fact that the limit of the product of two functions is equal to the product of their limits, assuming the limits exist.
Let's break down the problem step by step:
Step 1: Take the limit as x approaches infinity
We need to find the behavior of the function as x becomes increasingly large.
Step 2: Evaluate the expression sin(1/x)
Since the argument of the sine function, 1/x, approaches zero as x approaches infinity, we know that sin(1/x) will oscillate rapidly between -1 and 1. As x becomes larger, the oscillations become faster.
Step 3: Evaluate the final limit
To determine the overall behavior of the function, we need to multiply x by sin(1/x) and take the limit as x approaches infinity. Since sin(1/x) oscillates between -1 and 1, the product of x and sin(1/x) will oscillate between -x and x.
Since x is growing unbounded as x approaches infinity, the sine function will continue to oscillate between -1 and 1, but the amplitude of the oscillations will be magnified. As a result, the product of x and sin(1/x) does not approach a single number as x approaches infinity.
Therefore, the limit as x approaches positive infinity of x sin(1/x) does not exist.