By the definition of the derivative, this limit equals f'(4) when f(x) = 4/x^2. f(x)'s derivative, which can be easily calculated, exists at x = 4, so f(x) is differentiable at x = 4 and f'(4) exists, meaning that the limit exists. Alternatively, we can compute the limit algebraically by first reducing 4/16 to 1/4 to get:
and then since the LCD of the fractions in the numerator is 4(4 + h)^2, multiplying the numerator and denominator of the main fraction by 4(4 + h)^2 will clear the fractions in numerator. Doing this gives:
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By the definition of the derivative, this limit equals f'(4) when f(x) = 4/x^2. f(x)'s derivative, which can be easily calculated, exists at x = 4, so f(x) is differentiable at x = 4 and f'(4) exists, meaning that the limit exists. Alternatively, we can compute the limit algebraically by first reducing 4/16 to 1/4 to get:
lim (h-->0) [4/(4 + h)^2 - 4/16]/h = lim (h-->0) [4/(4 + h)^2 - 1/4]/h,
and then since the LCD of the fractions in the numerator is 4(4 + h)^2, multiplying the numerator and denominator of the main fraction by 4(4 + h)^2 will clear the fractions in numerator. Doing this gives:
lim (h-->0) [4/(4 + h)^2 - 4/16]/h = lim (h-->0) [4/(4 + h)^2 - 1/4]/h
= lim (h-->0) [4(4) - (4 + h)^2]/[4h(4 + h)^2]
= lim (h-->0) [16 - (h^2 + 8h + 16)]/[4h(4 + h)^2], by expanding the numerator
= -1/4 * lim (h-->0) (h^2 + 8h)/[h(4 + h)^2], by pulling out -1/4
= -1/4 * lim (h-->0) [h(h + 8)]/[h(4 + h)^2], by factoring the numerator
= -1/4 * lim (h-->0) (h + 8)/(4 + h)^2, by canceling the conflicting h factor
= (-1/4)(8/4^2), by evaluating the result at h = 0
= -1/8.
Thus, the limit exists and equals -1/8.