While proving that

MATHEMATICS · Single correct · CBSE · Mathematics · Limits and Derivatives · Limits and Derivatives

Problem

While proving that ddx[c f(x)]=c f′(x)\dfrac{d}{dx}[c\,f(x)]=c\,f'(x), the expression lim⁡h→0c f(x+h)−c f(x)h\displaystyle \lim_{h\to0}\dfrac{c\,f(x+h)-c\,f(x)}{h} is rewritten as c lim⁡h→0f(x+h)−f(x)hc\, \displaystyle \lim_{h\to0}\dfrac{f(x+h)-f(x)}{h}. Which limit property justifies moving cc outside the limit sign? (a) Product rule for limits (lim⁡h→0AhBh=(lim⁡h→0Ah)(lim⁡h→0Bh))\left(\displaystyle\lim_{h\to0}A_hB_h = \bigl(\lim_{h\to0}A_h\bigr)\bigl(\lim_{h\to0}B_h\bigr)\right) (b) Constant multiple rule for limits (lim⁡h→0cAh=c lim⁡h→0Ah)\left(\displaystyle \lim_{h\to0}cA_h = c\,\lim_{h\to0}A_h\right) (c) Continuity of ff at xx (d) Sandwich (Squeeze) theorem

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