QuestionCalculate the limit: $\lim _{t \rightarrow 5^{+}} \ln (t-5)$ .

Studdy Solution

STEP 1

Assumptions1. We are dealing with a limit problem. . The function is $\ln (t-5)$ .
3. We are finding the limit as $t$ approaches $5$ from the right, denoted as $5^{+}$ .

STEP 2

First, we need to understand what it means when we say $t$ is approaching $5$ from the right. This means that $t$ is getting very close to $5$ , but is slightly larger than $5$ .

STEP 3

Next, we need to understand what happens to the function $\ln (t-5)$ as $t$ gets very close to $5$ from the right. Since $t$ is slightly larger than $5$ , $t-5$ is a very small positive number.

STEP 4

The natural logarithm function, $\ln(x)$ , approaches negative infinity as $x$ approaches $0$ from the right. This is a property of the natural logarithm function.

STEP 5

Therefore, as $t$ approaches $5$ from the right, $\ln (t-5)$ approaches negative infinity.
$\lim{t \rightarrow5^{+}} \ln (t-5) = -\infty$ This means that the limit of $\ln (t-5)$ as $t$ approaches $5$ from the right is negative infinity.

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