How do you find the limit of # (1/(h+2)^2 - 1/4) / h# as h approaches 0?

Question

Scarlett Allison · Accepted Answer

We need first to manipulate the expression to put it in a more convenient form

Let's work on the expression #(1/(h+2)^2 -1/4)/h=((4-(h+2)^2)/(4(h+2)^2)) /h=((4-(h^2+4h+4))/(4(h+2)^2)) /h=(((4-h^2-4h-4))/(4(h+2)^2)) /h=(-h^2-4h)/(4(h+2)^2 h) = (h(-h-4))/(4(h+2)^2 h) = (-h-4)/(4(h+2)^2)# Taking now limits when #h-> 0# we have: #lim_(h->0)(-h-4)/(4(h+2)^2) = (-4)/16=-1/4#

Emma Aldridge · Answer

undefined To find the limit of the expression (1/(h+2)^2 - 1/4) / h as h approaches 0, we can simplify the expression first. By combining the fractions and finding a common denominator, we get ((4 - (h+2)^2) / (4(h+2)^2)) / h.

Next, we can simplify further by multiplying the numerator and denominator by h. This gives us (4h - h^3 - 4h - 8) / (4h(h+2)^2).

Simplifying the numerator, we have (-h^3 - 8) / (4h(h+2)^2).

Now, we can factor out a -1 from the numerator, resulting in -(h^3 + 8) / (4h(h+2)^2).

Since we are interested in finding the limit as h approaches 0, we can substitute 0 into the expression. Doing so, we get -(0^3 + 8) / (4(0)(0+2)^2), which simplifies to -8 / 0.

However, division by zero is undefined, so the limit does not exist in this case.