Consider the function
f(x)= 2e^x, if x<0
f(x) = -3x+2, if 0< or equal to x < or equal to 2/3
f(x) = 1, if x > or equal to 2/3
a) evaluate the limit X->0- and X->0+
b) Does limit X ->0 exist? Is f continuous at x=0
c) Evaluate the limit X->2/3- and X -> 2/3+
D) Does limit X->2/3 exist? Is f continuous at x=2/3?
To answer these questions, we will evaluate the limits by considering different parts of the given function based on the given conditions.
a) To evaluate the limit as x approaches 0- (from the left), we need to consider the function when x is less than 0. In this case, f(x) = 2e^x. Substituting x = 0 into this expression, we get f(0) = 2e^0 = 2(1) = 2.
To evaluate the limit as x approaches 0+ (from the right), we need to consider the function when x is greater than 0 but less than or equal to 2/3. In this case, f(x) = -3x + 2. Substituting x = 0 into this expression, we get f(0) = -3(0) + 2 = 2.
b) Since the limit as x approaches 0- is 2 and the limit as x approaches 0+ is also 2, we can conclude that the limit as x approaches 0 is 2. Hence, the limit X -> 0 exists.
To determine if f is continuous at x = 0, we need to check if the function approaches the same value from both sides. In this case, the left-sided limit (as x approaches 0-) is 2, and the right-sided limit (as x approaches 0+) is also 2. Since these two limits are equal, we can conclude that f is continuous at x = 0.
c) To evaluate the limit as x approaches 2/3- (from the left), we need to consider the function when x is less than 2/3. In this case, f(x) = -3x + 2. Substituting x = 2/3 into this expression, we get f(2/3) = -3(2/3) + 2 = 2/3.
To evaluate the limit as x approaches 2/3+ (from the right), we need to consider the function when x is greater than or equal to 2/3. In this case, f(x) = 1. Substituting x = 2/3 into this expression, we get f(2/3) = 1.
d) Since the limit as x approaches 2/3- is 2/3, and the limit as x approaches 2/3+ is 1, we can conclude that the limit as x approaches 2/3 does not exist. The left-sided and right-sided limits are not equal.
To determine if f is continuous at x = 2/3, we need the left-sided and right-sided limits to equal the function value at x = 2/3. However, since the limit as x approaches 2/3 does not exist, we can conclude that f is not continuous at x = 2/3.