Find the Limits of f(x)=(x^2-16)/(sqrt(x^2-8x+16)) at x=4

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In summary, the speaker is asking for help with finding the left, right, and overall limit of a given function. They have attempted to simplify the problem but are unsure of what to do next. They provide their work and state that the function is piecewise, making it easy to find the left and right limit. They also suggest graphing the function as an alternative method.
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chukie
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Hi I was wondering if someone could help me with this limit problem:

Find the lim x->a-, lim x->a+ and lim x->a
for the f(x)=(x^2-16)/(squareroot(x^2-8x+16)) a=4

I've tried to simplify the problem but after that i don't know wut to do =(
This is wut I've done:
y=(x^2-16)/(squareroot(x-4)^2)
y=(x^2-16)/abs(x-4)

any help would be appreciated.
 
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  • #2
y=(x^2-16)/abs(x-4)
= x+4,x>4
=-x-4,x<4
this is a piece wise function, mow you can easily find the left,right limit
Note:Limit of f(x) as x approaches 4 doesn't exist as left and right limit at x=4 are not equal

The more intuitive way is to draw the graph of the piecewise function!
 

1. What is the limit of the function at x=4?

The limit of the function f(x)=(x^2-16)/(sqrt(x^2-8x+16)) at x=4 is undefined. This is because at x=4, the denominator of the function becomes 0, which results in an indeterminate form.

2. How can I find the limit algebraically?

To find the limit algebraically, you can use the rationalization method. This involves multiplying the numerator and denominator of the function by the conjugate of the denominator, which in this case is sqrt(x^2-8x+16)+4. This will eliminate the square root in the denominator and allow you to simplify the expression and find the limit.

3. Can I use a graph to find the limit?

Yes, you can use a graph to estimate the limit of the function. By plotting the function on a graphing calculator or software, you can see that as x approaches 4 from both the left and right sides, the function approaches positive infinity. This indicates that the limit at x=4 is undefined.

4. What happens if I approach x=4 from the left or right side?

If you approach x=4 from the left side, the function will approach positive infinity. This is because as x gets closer to 4 from the left side, the denominator becomes a very small positive number, causing the overall fraction to become a large positive value. If you approach x=4 from the right side, the function will also approach positive infinity for the same reason.

5. How does the domain of the function relate to the limit at x=4?

The fact that the denominator of the function becomes 0 at x=4 indicates that x=4 is not in the domain of the function. This means that the function is not defined at x=4, and therefore the limit cannot exist at that point.

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