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Suppose that we know the reciprocal rule for limits: If limxcg(x)=Mexists and is nonzero, then limxc1g(x)=1MThis limit rule is tedious to prove and we do not include it here. Use the reciprocal rule and the product rule for limits to prove the quotient rule for limits.

Short Answer

Expert verified

Ans: limxcf(x)g(x)=limxcf(x)limxcg(x)

Step by step solution

01

Step 1. Given Information:

The reciprocal rule is given by,

if limxcg(x)=Mexists and is nonzero, thenlimxc1g(x)=1M

02

Step 2. Prove:

If limxcf(x)exists and limxcg(x)exists and is nonzero, then by the product and reciprocal rules for limit we have,

localid="1668094027457" limxcf(x)g(x)=limxcf(x)1g(x)=limxcf(x)limxc1g(x)=limxcf(x)1limxcg(x)=limxcf(x)limxcg(x)

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