F(X)= x-2/(x^2-4) is continuous at x=1 and why as limits approach from 1+ and 1-
why what?
limit as x -> 1 from either side is
(-1-2)/(1-4) = -3/-3 = 1
one would expect this, as f is continuous.
oops. I did x -> -1
limit as x -> 1 is
(1-2)/(1-4) = 1/3
Makes sense, since f(x) = 1/(x+2) for x ≠ ±2
To determine if a function is continuous at a particular value, such as x = 1 in this case, we need to check if the left-hand limit (as x approaches the value from the left) and the right-hand limit (as x approaches the value from the right) both exist and are equal to each other.
Let's start by finding the left-hand limit as x approaches 1- (from the left side). We substitute values of x that are less than 1 into the function f(x) = (x - 2) / (x^2 - 4):
lim x→1- (x - 2) / (x^2 - 4)
Now, substituting x = 1 - h, where h is a small positive value approaching zero:
lim h→0- ((1 - h) - 2) / ((1 - h)^2 - 4)
Next, we simplify the expression:
lim h→0- (-h - 1) / (1 - 2h + h^2 - 4)
Now, we can calculate the limit:
lim h→0- (-h - 1) / (h^2 - 2h - 3)
Using algebraic manipulation, we factor the denominator:
lim h→0- (-h - 1) / [(h - 3)(h + 1)]
Now, it is clear that the denominator is nonzero for a small value of h. Therefore, the limit exists and is finite. Evaluating the limit with direct substitution (plug in h = 0) gives:
lim h→0- (-0 - 1) / [(0 - 3)(0 + 1)] = -1 / (-3) = 1/3
Now, let's find the right-hand limit as x approaches 1+ (from the right side). We substitute values of x that are greater than 1 into the function f(x):
lim x→1+ (x - 2) / (x^2 - 4)
Substituting x = 1 + h, where h is a small positive value approaching zero:
lim h→0+ ((1 + h) - 2) / ((1 + h)^2 - 4)
Simplifying the expression:
lim h→0+ (h - 1) / (1 + 2h + h^2 - 4)
Combining like terms:
lim h→0+ (h - 1) / (h^2 + 2h - 3)
Factoring the denominator:
lim h→0+ (h - 1) / [(h - 1)(h + 3)]
Now, it is clear that the denominator is nonzero for a small value of h. Therefore, the limit exists and is finite. Evaluating the limit with direct substitution (plug in h = 0) gives:
lim h→0+ (0 - 1) / [(0 - 1)(0 + 3)] = -1 / (-3) = 1/3
Since the left-hand limit and the right-hand limit are both equal to 1/3, we can conclude that the function f(x) = (x - 2) / (x^2 - 4) is continuous at x = 1.