A piecewise function is given. Use the function to find the indicated limits, or state that a limit does not exist.
(a) lim is over x-->d – f(x), (b) lim is over x-->d + f(x), and (c) lim is over x-->d f(x)
x^2 - 5 if x < 0
f(x)= -2 if x ->0 ; d = -3
1. (a)-5 (b)-2 (c)does not exist
2. (a)4 (b)4 (c)4
3. (a)-2 (b)-5 (c)does not exist
4. (a)-5 (b)-2 (c)-2
To find the limits of the given piecewise function, you need to evaluate the function as x approaches the specified value (d) from the left (a), from the right (b), and at the actual value (c) itself. Here's how you can find the limits:
(a) lim as x approaches d- (from the left):
To find this limit, you need to evaluate f(x) for values of x that are slightly less than d (or less than d if d is included in the domain). In this case, d = -3. So, you would evaluate f(x) for values of x that are slightly to the left (less than) -3. Since x^2 - 5 holds for x < 0, as you approach -3 from the left, you use the expression x^2 - 5. Plugging in -3 into this expression gives (-3)^2 - 5 = 4 - 5 = -1. Therefore, the limit as x approaches -3 from the left is -1.
(b) lim as x approaches d+ (from the right):
To find this limit, you need to evaluate f(x) for values of x that are slightly greater than d (or greater than d if d is included in the domain). In this case, d = -3. So, you would evaluate f(x) for values of x that are slightly to the right (greater than) -3. Since -2 is explicitly defined for x = 0, and there is no other expression for x > 0, the value of f(x) as x approaches -3 from the right would be -2. Therefore, the limit as x approaches -3 from the right is -2.
(c) lim as x approaches d:
To find this limit, you need to evaluate the function at the actual value of d. In this case, d = -3. Since x^2 - 5 is only defined for x < 0, and -2 is explicitly defined for x = 0, there is no expression defined for x = -3. Therefore, the limit as x approaches -3 does not exist.
Based on the above calculations, the correct answer is:
(a) lim as x approaches -3- (from the left) is -1;
(b) lim as x approaches -3+ (from the right) is -2;
(c) The limit as x approaches -3 does not exist.