The graph of h(x) is shown. The graph appears to be continuous at , where the definition changes. Show that h is in fact continuous at by finding the left and right limits and showing that the definition of continuity is met?

Answer 1

Kindly refer to the Explanation.

To show that #h# is continuous, we need to check its
continuity at #x=3#.
We know that, #h# will be cont. at #x=3#, if and only if,
#lim_(x to 3-) h(x)=h(3)=lim_(x to 3+) h(x)...............................(ast)#.
As #x to 3-, x lt 3 :. h(x)=-x^2+4x+1#.
#:. lim_(x to 3-)h(x)=lim_(x to 3-)-x^2+4x+1=-(3)^2+4(3)+1#,
# rArr lim_(x to 3-)h(x)=4......................................................(ast^1)#.
Similarly, #lim_(x to 3+)h(x)=lim_(x to 3+)4(0.6)^(x-3)=4(0.6)^0#.
# rArr lim_(x to 3+)h(x)=4....................................................(ast^2)#.
Finally, #h(3)=4(0.6)^(3-3)=4......................................(ast^3)#.
#(ast),(ast^1),(ast^2) and (ast^3) rArr h" is cont. at "x=3#.
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Answer 2

See below:

For a function to be continuous at a point (call it 'c'), the following must be true:

#f(c)# must exist.
#lim_(x->c)f(x)# must exist

The former is defined to be true, but we'll need to verify the latter. How? Well, recall that for a limit to exist, the right and left hand limits must equal the same value. Mathematically:

#lim_(x->c^-)f(x) = lim_(x->c^+)f(x)#

This is what we'll need to verify:

#lim_(x->3^-)f(x) = lim_(x->3^+)f(x)#
To the left of #x = 3#, we can see that #f(x) = -x^2 +4x + 1#. Also, to the right of (and at) #x = 3#, #f(x) = 4(0.6^(x-3))#. Using this:
#lim_(x->3)-x^2 +4x + 1 = lim_(x->3)4(0.6^(x-3))#

Now, we just evaluate these limits, and check if they're equal:

#-(3^2) + 4(3) + 1 = 4(0.6^(3-3))#
#=> -9 + 12 + 1 = 4(0.6^0)#
#=> 4 = 4#
So, we have verified that #f(x)# is continuous at #x = 3#.

Hope that helped :)

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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