# How do you prove that the limit of #3x+5=35# as x approaches 10 using the precise definition of a limit?

Functions of this kind are caled Lipschitz continuous functions.

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To prove that the limit of 3x+5=35 as x approaches 10 using the precise definition of a limit, we need to show that for any positive value ε, there exists a positive value δ such that if 0 < |x - 10| < δ, then |(3x+5) - 35| < ε.

Let's start by simplifying the expression (3x+5) - 35 = 3x - 30. We want to find a δ that ensures |3x - 30| < ε whenever 0 < |x - 10| < δ.

Now, let's manipulate the inequality |3x - 30| < ε to isolate x. We divide both sides by 3: |x - 10| < ε/3.

From this, we can see that if we choose δ = ε/3, then whenever 0 < |x - 10| < δ, we have |(3x+5) - 35| < ε.

Therefore, by choosing δ = ε/3, we have proven that the limit of 3x+5=35 as x approaches 10 using the precise definition of a limit.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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