Limit Problem

[Jason76]

Given: \(\displaystyle g(x) = x^{2} + 5\) if \(\displaystyle x < - 2\) and \(\displaystyle 1 - 3x\) if \(\displaystyle x \geq -2\)

(a) \(\displaystyle \lim x \rightarrow 6 = 1 - 3(6) = -17\)

(b) \(\displaystyle \lim x \rightarrow -2\)

\(\displaystyle \lim x \rightarrow 2- = (-2)^{2} + 5 = 9\)

\(\displaystyle \lim x \rightarrow 2+ = (1 - 3(-2)) = 7\)

\(\displaystyle \lim x \rightarrow -2\) DNE (limits from the left and the right don't match)

How does (a) differ from (b)?b I totally understand the reasoning, but not a. How did we know in (a), which equation to find the limit?

 

[stapel]

Given: \(\displaystyle g(x) = x^{2} + 5\) if \(\displaystyle x < - 2\) and \(\displaystyle 1 - 3x\) if \(\displaystyle x \geq -2\)

(a) \(\displaystyle \lim x \rightarrow 6 = 1 - 3(6) = -17\)

How did we know in (a), which equation to find the limit?

Look at the definition for the function. For x = 6, which piece of the function is relevant?

 

[HallsofIvy]

Given: \(\displaystyle g(x) = x^{2} + 5\) if \(\displaystyle x < - 2\) and \(\displaystyle 1 - 3x\) if \(\displaystyle x \geq -2\)

(a) \(\displaystyle \lim x \rightarrow 6 = 1 - 3(6) = -17\)

(b) \(\displaystyle \lim x \rightarrow -2\)

\(\displaystyle \lim x \rightarrow 2- = (-2)^{2} + 5 = 9\)

\(\displaystyle \lim x \rightarrow 2+ = (1 - 3(-2)) = 7\)

\(\displaystyle \lim x \rightarrow -2\) DNE (limits from the left and the right don't match)

How does (a) differ from (b)?b I totally understand the reasoning, but not a. How did we know in (a), which equation to find the limit?

Because we know what the word "if" means! We are taking the limit as x goes to 6 so we are only concerned with what happens for x close to 6. "Close to 6" is certainly "larger than -2". What happens for x less than -2 is not relevant.

But in (b) we are taking the limit as x goes to -2. We have to consider values of x on both sides of -2.