Questions: If the limit of f(x) as x approaches 5 from the right is greater than 0 and the limit of f(x) as x approaches 5 from the left is greater than 0, then the limit of f(x) as x approaches 5 is greater than 0.

If the limit of f(x) as x approaches 5 from the right is greater than 0 and the limit of f(x) as x approaches 5 from the left is greater than 0, then the limit of f(x) as x approaches 5 is greater than 0.

Transcript text: If $\lim _{x \rightarrow 5^{+}} f(x)>0$ and $\lim _{x \rightarrow 5^{-}} f(x)>0$, then $\lim _{x \rightarrow 5} f(x)>0$.

Solution

Solution Steps

To determine if $\lim _{x \rightarrow 5} f(x) > 0$, we need to check if both the right-hand limit $\lim _{x \rightarrow 5^{+}} f(x)$ and the left-hand limit $\lim _{x \rightarrow 5^{-}} f(x)$ are greater than zero. If both are greater than zero, then the two-sided limit $\lim _{x \rightarrow 5} f(x)$ is also greater than zero.

Step 1: Evaluate the Right-Hand Limit

Given that $\lim _{x \rightarrow 5^{+}} f(x) = 0.1$, we observe that this limit is greater than zero.

Step 2: Evaluate the Left-Hand Limit

Given that $\lim _{x \rightarrow 5^{-}} f(x) = 0.2$, we observe that this limit is also greater than zero.

Step 3: Determine the Two-Sided Limit

Since both the right-hand limit and the left-hand limit are greater than zero, we conclude that the two-sided limit $\lim _{x \rightarrow 5} f(x)$ is also greater than zero.

Final Answer

\[ \boxed{\lim _{x \rightarrow 5} f(x) > 0} \]

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