The left hand derivative of f x x sin pi x at x k Where k is an integer and x is the ...........

Mathematics (NDA) limits

The left-hand derivative of f(x) = [x] sin(πx) at x = k

Where k is an integer and [x] is the greatest integer function, is-

(−1)^k (k − 1)π

(−1)^k-1 (k − 1)π

(−1)^kkπ

(−1)^k-1 kπ

LHD = lim f(k)−f(k−h) / h
h→0

= lim k sinkπ−(k−h)sin(k−h)π / h
h→0

= lim −(k−1) sin(k−h)π / h
h→0

sin kπ = 0

Sin (kπ – πh) = (-k)^k-1 sin hπ

=lim −(k−1)−(−1)^k-1 sin hπ x π / hπ
h→0

= (-1)^k (k-1) π

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