Limit Find K?

[ervinako]

QUESTION:
Let
h(x){ 1+x³/1+x if x<-1
h(x){ 3 if x = -1
h(x){ 2x+5/k if x > -1


find k so that lim h(x) exists.

• My problem is how to determine where X is to be approach?

like X -> 1? or what? , is this X -> to -1?

• The other example that our professor give is like this:

f(x) {2k+5x if 0<x<1. k-3x if 1<x<2. Where X -> 1.

• I know that x<-1 is the X -> a^- and X -> a⁺ is the x >-1.

and the x= -1 is the two sided limit.

• We all know that lim 3 = lim 1+x³ / x+1 = lim 2x+5/k.

 

[JeffM]

QUESTION:
Let
h(x){ 1+x³/1+x if x<-1 Is this supposed to be h(x) = (1 + x^3) / (1 + x) if x < - 1?
h(x){ 3 if x = -1
h(x){ 2x+5/k if x > -1 Is this supposed to be h(x) = (2x + 5) / k if x > - 1?


find k so that lim h(x) exists.

• My problem is how to determine where X is to be approach?

like X -> 1? or what? , is this X -> to -1

\(\displaystyle \displaystyle \lim_{x \rightarrow a}h(x)\) obviously exists if a < - 1 or if a > 1.

So the issue is the limit when a = - 1. But let's take it in small steps.

Please answer the following questions.

\(\displaystyle a < - 1 \implies \displaystyle \lim_{x \rightarrow a}h(x) = ?\)

That should have been very easy.

\(\displaystyle a > - 1 \implies \displaystyle \lim_{x \rightarrow a}h(x) = ?\)

That too should have been very easy.

\(\displaystyle \displaystyle \lim_{x \rightarrow a^+}h(x) = ?\)

Presumably no problems so far.

\(\displaystyle \displaystyle \lim_{x \rightarrow a^-}h(x) = ?\)

What do you get for this one?

Now \(\displaystyle \displaystyle \lim_{x \rightarrow a^-}h(x) = \lim_{x \rightarrow a^+}h(x) \implies \lim_{x \rightarrow a}h(x) = \lim_{x \rightarrow a^-}h(x) = \lim_{x \rightarrow a^+}h(x)\)

The limit exists only if the right and left limits exist and are equal.

What value of k achieves that result?