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)#

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Answer 1 · 2017-12-13T20:18:39

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)