Prove the constant multiple rule for limits:limx→cf(x)=L and k∈ℝ, then limx→ckf(x)=kL

Ans: if x∈(c-δ,c)∪(c,c+δ),then f(x) is within ε|k| of L

Q. 91 - Calculus | StudyQuestionHub

Q. 91

Question

Prove the constant multiple rule for limits:
$\underset{x \to c}{l i m} f (x) = L and k \in ℝ, then \underset{x \to c}{l i m} k f (x) = k L$

Step-by-Step Solution

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Answer

Ans: $i f x \in (c - δ, c) \cup (c, c + δ), t h e n f (x) i s w i t h i n \frac{ε}{| k |} o f L$

1Step 1. Given Information:

$\lim_{x \to c} f (x) = L and k \in ℝ, then \lim_{x \to c} k f (x) = k L$

2Step 2. Prove:

The strategy is to prove the constant multiple rules for limits, considering that

$\lim_{x \to c} f (x) = L and k \in ℝ, then \lim_{x \to c} k f (x) = k L$

3Step 3.Constant multiple rule for limits:

$G i e r n ε > 0, w e c a n c h o o s e δ > 0 s o t h a t i f x \in (c - δ, c) \cup (c, c + δ), t h e n f (x) i s w i t h i n \frac{ε}{|k|} o f L . |k f (x) - k L| = | k (f (x) - L) = | k | | f (x) - L | < | k | δ = | k | (\frac{ε}{| k |}) = ε H e n c e, i f \lim_{x \to c} f (x) = L and k \in ℝ, then \lim_{x \to c} k f (x) = k L$

Q. 90

Q. 92

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