To find the left-hand limit of the function [tex]\( f(x) = \sqrt{x^2 - 4} \)[/tex] as [tex]\( x \)[/tex] approaches 2, we need to carefully analyze the behavior of the function as [tex]\( x \)[/tex] gets closer to 2 from the left side ([tex]\( x \to 2^- \)[/tex]).
1. Understand the Function:
The function [tex]\( f(x) = \sqrt{x^2 - 4} \)[/tex] is defined for [tex]\( x \)[/tex] such that [tex]\( x^2 - 4 \geq 0 \)[/tex] because the square root function requires a non-negative argument.
[tex]\[
\sqrt{x^2 - 4} \text{ is defined when } x^2 - 4 \geq 0
\][/tex]
Solving [tex]\( x^2 - 4 \geq 0 \)[/tex]:
[tex]\[
x^2 \geq 4
\][/tex]
[tex]\[
|x| \geq 2
\][/tex]
This means the function [tex]\( f(x) \)[/tex] is defined for [tex]\( x \leq -2 \)[/tex] or [tex]\( x \geq 2 \)[/tex].
2. Evaluate the Limit as [tex]\( x \)[/tex] Approaches 2 from the Left:
We are interested in the behavior of [tex]\( f(x) \)[/tex] as [tex]\( x \)[/tex] approaches 2 from the left ([tex]\( x \to 2^- \)[/tex]). This implies [tex]\( 2 > x > -\infty \)[/tex].
Let's substitute [tex]\( x \)[/tex] with values close to 2 but less than 2 (like 1.9, 1.99, etc.).
3. Rewrite the Expression:
For values of [tex]\( x \)[/tex] close to 2 but less than 2:
[tex]\[
f(x) = \sqrt{x^2 - 4}
\][/tex]
As [tex]\( x \)[/tex] approaches 2 from the left, we have [tex]\( x \)[/tex] slightly less than 2.
Let's consider the behavior of [tex]\( x^2 - 4 \)[/tex]:
When [tex]\( x = 2 \)[/tex]:
[tex]\[
x^2 - 4 = 2^2 - 4 = 4 - 4 = 0
\][/tex]
When [tex]\( x \)[/tex] is slightly less than 2 but very close to it (say [tex]\( x = 2 - \epsilon \)[/tex], for very small [tex]\( \epsilon \)[/tex]):
[tex]\[
\left(2 - \epsilon\right)^2 - 4 \approx 0
\][/tex]
Here, [tex]\( \epsilon \)[/tex] is a small positive number. As [tex]\( x \)[/tex] approaches 2 from the left, the term [tex]\( x^2 - 4 \approx 0 \)[/tex] and becomes very small but still non-negative.
4. Limit Calculation:
[tex]\[
\lim_{x \to 2^-} \sqrt{x^2 - 4} = \sqrt{0} = 0
\][/tex]
Thus, the left-hand limit of the function [tex]\( f(x) = \sqrt{x^2 - 4} \)[/tex] as [tex]\( x \)[/tex] approaches 2 is:
[tex]\[
\lim_{x \to 2^-} \sqrt{x^2 - 4} = 0
\][/tex]
