Use the definition of continuity and the properties of limits to show that the function is continuous at the given number a.
f(x)= (x+2x^3)^4, a=1.
PLEASE HELP ME WITH THIS PROBLEM. THANK YOU!
f(x)= (x+2x^3)^4, a=1.
PLEASE HELP ME WITH THIS PROBLEM. THANK YOU!

If f(x) is continuous at x = a, then:
lim (x>a) f(x) = lim (x>a+) f(x) = f(a).
Since:
(i) lim (x>1) f(x)
= lim (x>1) (x + 2x^3)^4
= [1 + 2(1)]^4
= 81
(ii) lim (x>1+) f(x)
= lim (x>1+) (x + 2x^3)^4
= [1 + 2(1)]^4
= 81
(iii) f(1) = [1 + 2(1)]^4 = 81
==> lim (x>1) f(x) = lim (x>1+) f(x) = f(1),
we see that f(x) is continuous at x = 1.
I hope this helps!
lim (x>a) f(x) = lim (x>a+) f(x) = f(a).
Since:
(i) lim (x>1) f(x)
= lim (x>1) (x + 2x^3)^4
= [1 + 2(1)]^4
= 81
(ii) lim (x>1+) f(x)
= lim (x>1+) (x + 2x^3)^4
= [1 + 2(1)]^4
= 81
(iii) f(1) = [1 + 2(1)]^4 = 81
==> lim (x>1) f(x) = lim (x>1+) f(x) = f(1),
we see that f(x) is continuous at x = 1.
I hope this helps!

the limit does not exist