Evaluate the limit. $$ \lim _{x ightarrow \frac{\pi}{2}^{-}} \frac{ an x}{\left(\frac{20}{2 x-\pi} ight)}=\square $$ (Type an exact answer.)

Question

Evaluate the limit. $$ \lim _{x ightarrow \frac{\pi}{2}^{-}} \frac{	an x}{\left(\frac{20}{2 x-\pi}ight)}=\square $$ (Type an exact answer.)

UpStudy AI · Accepted Answer

To evaluate the limit:

$$
\lim _{x \rightarrow \frac{\pi}{2}^{-}} \frac{\tan x}{\left(\frac{20}{2 x-\pi}\right)}
$$

Let's simplify the expression step by step.

1. **Rewrite the Limit:**

$$
\lim _{x \rightarrow \frac{\pi}{2}^{-}} \frac{\tan x}{\frac{20}{2x - \pi}} = \lim _{x \rightarrow \frac{\pi}{2}^{-}} \tan x \cdot \frac{2x - \pi}{20}
$$

2. **Substitution:**

Let $$ t = \frac{\pi}{2} - x $$. As $$ x \rightarrow \frac{\pi}{2}^{-} $$, $$ t \rightarrow 0^{+} $$.

Express the terms in terms of $$ t $$:

$$
x = \frac{\pi}{2} - t \
2x - \pi = 2\left(\frac{\pi}{2} - t\right) - \pi = -2t \
\tan x = \tan\left(\frac{\pi}{2} - t\right) = \cot t = \frac{1}{\tan t}
$$

3. **Substitute Back into the Limit:**

$$
\tan x \cdot \frac{2x - \pi}{20} = \frac{1}{\tan t} \cdot \frac{-2t}{20} = \frac{-2t}{20 \tan t} = \frac{-t}{10 \tan t}
$$

4. **Evaluate the Limit:**

$$
\lim_{t \rightarrow 0^{+}} \frac{-t}{10 \tan t}
$$

We know that:

$$
\lim_{t \rightarrow 0} \frac{\tan t}{t} = 1 \implies \lim_{t \rightarrow 0} \frac{t}{\tan t} = 1
$$

Therefore:

$$
\lim_{t \rightarrow 0^{+}} \frac{-t}{10 \tan t} = \frac{-1}{10}
$$

**Final Answer:**

$$
- \frac{1}{10}
$$
The limit is $$ -\frac{1}{10} $$.

Evaluate the limit.
(Type an exact answer.)

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