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Question
Evaluate the following :
If f(r) = πr2 then find `lim_("h" -> 0) [("f"("r" + "h") - "f"("r"))/"h"]`
Solution
f(r) = πr2
f(r + h) = π(r + h)2
= π(r2 + 2rh + h2)
= πr2 + 2πrh + πh2
∴ `lim_("h" -> 0) ("f"("r" + "h") - "f"("r"))/"h"`
= `lim_("h" -> 0) (pi"r"^2 + 2pi"rh" + pi"h"^2 - pi"r"^2)/"h"`
= `lim_("h" -> 0) (2pi"rh" + pi"h"^2)/"h"`
= `lim_("h" -> 0) ("h"(2pi"h" + pi"h"))/"h"`
= `lim_("h" -> 0) (2pi"r" + pi"h")` ...[∵ h → 0, ∵ h ≠ 0]
= 2πr + π(0)
= 2πr
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