How do you find the limit of ((1/x+16)-(1/16)) / x(1x+16)−(116)x as x approaches 0?
1 Answer
I think you meant:
When we try to find the limit, we get the (indeterminate) form
The important algebra can be done in a couple ways:
Method 1
"If I had a fraction over a fraction, I'd know what to do next."
Good! Make it so.
= ((16- (x+16))/(16(x+16)))/(x/1)=16−(x+16)16(x+16)x1
= (-x)/(16(x+16))*1/x=−x16(x+16)⋅1x
= (-1)/(16(x+16))=−116(x+16)
Method 2
"I know this trick:"
Multiply numerator and denominator by the common denominator of all the fractions in the numerator and denominator. (Sounds complicated, but it's quicker.)
The common denominator is
= ((16(x+16))/(x+16) - (16(x+16))/16)/(x(16(x+16))=16(x+16)x+16−16(x+16)16x(16(x+16))
=(16-(x+16))/(16x(x+16))=16−(x+16)16x(x+16)
= (-1)/(16(x+16))=−116(x+16)
So, we get: