Suppose #lim_(x→0
+)f(x) = A and lim_(x→0
-)f(x) = B.# Determine
#a. lim_(x→0−)
f(x^2 − x)#
#b. lim_(x→0−)(f(x^2) − f(x))#
#c. lim_(x→0+)f(x^3 − x)#
#d. lim_(x→0−)(f(x^3) − f(x))#
#e. lim_(x→1−)f(x^2 − x)#?

Question

Suppose #lim_(x→0 +)f(x) = A and lim_(x→0 -)f(x) = B.# Determine
#a. lim_(x→0−) f(x^2 − x)#
#b. lim_(x→0−)(f(x^2) − f(x))#
#c. lim_(x→0+)f(x^3 − x)#
#d. lim_(x→0−)(f(x^3) − f(x))#
#e. lim_(x→1−)f(x^2 − x)#

Christopher Agresta · Accepted Answer

These are nice questions!

(a) As #xrarr0^-#, #x^2-x = x(x-1)# is a product of two negative numbers, so #x(x-1) rarr0^+# and #lim_(xrarr0^-)f(x^2-x) = lim_(xrarr0^+)f(x) = A# (b) As #xrarr0^-#, #x^2 rarr0^+# so #lim_(xrarr0^-)(f(x^2)-f(x)) = A-B# (c) use #x^3-x = x(x^2-1)# to see that the limit is #B# (d)#0# (e)

Emilia Aguilar · Answer

undefined a. lim_(x→0−) f(x^2 − x) = lim_(x→0−) f(x^2 − x) = B

b. lim_(x→0−) (f(x^2) − f(x)) = lim_(x→0−) (f(x^2) − f(x)) = B - A

c. lim_(x→0+) f(x^3 − x) = lim_(x→0+) f(x^3 − x) = A

d. lim_(x→0−) (f(x^3) − f(x)) = lim_(x→0−) (f(x^3) − f(x)) = A - B

e. lim_(x→1−) f(x^2 − x) = lim_(x→1−) f(x^2 − x) = A

HIX Tutor · Answer

undefined a. To determine lim_(x→0-) f(x^2 - x), substitute x^2 - x into f(x) using the definition of the limit. This gives lim_(x→0-) f(x^2 - x) = lim_(x→0-) f(0) = B.

b. To find lim_(x→0-) (f(x^2) - f(x)), substitute x^2 into f(x) and then subtract f(x) from it. This equals lim_(x→0-) (f(x^2) - f(x)) = lim_(x→0-) f(0) - lim_(x→0-) f(0) = 0.

c. For lim_(x→0+) f(x^3 - x), substitute x^3 - x into f(x). This results in lim_(x→0+) f(x^3 - x) = f(0) = A.

d. To calculate lim_(x→0-) (f(x^3) - f(x)), substitute x^3 into f(x) and then subtract f(x) from it. This simplifies to lim_(x→0-) (f(x^3) - f(x)) = lim_(x→0-) f(0) - lim_(x→0-) f(0) = 0.

e. To determine lim_(x→1-) f(x^2 - x), substitute x^2 - x into f(x). Since the function's behavior near x = 1 is not given, we cannot determine this limit with the provided information.

Suppose #lim_(x→0 +)f(x) = A and lim_(x→0 -)f(x) = B.# Determine #a. lim_(x→0−) f(x^2 − x)# #b. lim_(x→0−)(f(x^2) − f(x))# #c. lim_(x→0+)f(x^3 − x)# #d. lim_(x→0−)(f(x^3) − f(x))# #e. lim_(x→1−)f(x^2 − x)#?

Suppose #lim_(x→0 +)f(x) = A and lim_(x→0 -)f(x) = B.# Determine #a. lim_(x→0−) f(x^2 − x)# #b. lim_(x→0−)(f(x^2) − f(x))# #c. lim_(x→0+)f(x^3 − x)# #d. lim_(x→0−)(f(x^3) − f(x))# #e. lim_(x→1−)f(x^2 − x)#

Suppose #lim_(x→0 +)f(x) = A and lim_(x→0 -)f(x) = B.# Determine
#a. lim_(x→0−) f(x^2 − x)#
#b. lim_(x→0−)(f(x^2) − f(x))#
#c. lim_(x→0+)f(x^3 − x)#
#d. lim_(x→0−)(f(x^3) − f(x))#
#e. lim_(x→1−)f(x^2 − x)#