Chapter 1: Q. 91 (page 137)

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

Short Answer

Expert verified

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

Step by step solution

Step 1. Given Information:

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

Step 2. Prove:

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

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

Step 3.Constant multiple rule for limits:

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

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

Short Answer

Step by step solution

Step 1. Given Information:

Step 2. Prove:

Step 3.Constant multiple rule for limits:

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Prove the constant multiple rule for limits:limx→cf(x)=Landk∈ℝ,thenlimx→ckf(x)=kL

Short Answer

Step by step solution

Step 1. Given Information:

Step 2. Prove:

Step 3.Constant multiple rule for limits:

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

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Study anywhere. Anytime. Across all devices.

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