How do you use the definition of continuity and the properties of limits to show that the function #f(x) = 5x^4 - 9x^3 + x - 7# is continuous at a given number a=7?

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Answer 1 · 2016-02-25T22:30:32

#lim_(xrarr7)f(x) = lim_(xrarr7)(5x^4 - 9x^3 + x - 7)#

# = lim_(xrarr7)(5x^4) - lim_(xrarr7) (9x^3) +lim_(xrarr7) (x) -lim_(xrarr7) (7)#

(Limit of a sum and difference)

# = 5lim_(xrarr7)(x^4) - 9lim_(xrarr7) (x^3) +lim_(xrarr7) (x) -lim_(xrarr7) (7)#

(Limit of a constant multiple of a function)

# = 5(lim_(xrarr7)x)^4 - 9(lim_(xrarr7) x)^3 +lim_(xrarr7) (x) -lim_(xrarr7) (7)#

(Limit of a power or repeated limits of products.)

# = 5(7)^4-9(7)^3+(7)-(7)#

(Limits of identity and constant functions)

# = f(7)#

We have shown that #lim_(xrarr7)f(x) = f(7)#, so, by the definition of continuity (at a point), #f# is continuous at #7#