Find each of the limits without a calculator. Use the limits as an aide to sketch the graph of f(x)= (x-2)/(|x|-2) for-10<_x<_10 and -6<_y<_6. Be sure to clearly indicate any asymptotes, holes or other important characteristics.
lim x->∞ f(x)
recall the definition of |u|. It is
u for u>=0, and -u for u<0. So, your function
f(x) =
(x-2)/(x-2) for x >= 2
(x-2)/(-x-2) for x < 2
Now you can easily find the necessary limits. Then, peek at the graph below to verify your calculations.
Would I do the same thing as x approaches:
Infinity
-2+
-2-
2+
2-
To find the limit as x approaches infinity for the function f(x) = (x-2)/(|x|-2), we need to analyze the behavior of the function as x becomes larger and larger.
In this case, as x approaches infinity, the denominator |x| - 2 will also approach infinity. Since the denominator grows larger without bound, we can consider only the term x-2 in the numerator.
The limit of f(x) as x approaches infinity can be expressed as:
lim x->∞ (x-2)/(|x|-2) = lim x->∞ (x-2)/x
To simplify further, we can divide both the numerator and denominator by x:
lim x->∞ (x/x - 2/x) = lim x->∞ (1 - 2/x) = 1 - 0 = 1
Hence, the limit of f(x) as x approaches infinity is 1.
To sketch the graph of f(x), we can start by plotting some key points:
- When x = 0, the function is undefined since it involves division by zero. This indicates a vertical asymptote at x = 0.
- When x = 2, the function is also undefined since it involves division by zero. This indicates a hole in the graph at x = 2.
- As x approaches -2 from the left, the function approaches negative infinity. This implies a vertical asymptote at x = -2.
- As x approaches -2 from the right, the function approaches positive infinity. This also implies a vertical asymptote at x = -2.
Using these points and the limit we found earlier, we can sketch the graph of f(x) within the specified domain. Remember to consider the characteristics we mentioned above: asymptotes, holes, and other important points.