Limit as x approaches (pi/2)^- of (x-pi/2)*secx
(x-π/2) secx
since cosx = sin(π/2-x), we have
-(π/2-x)/sin(π/2-x)
that's just like -u/sinu -> -1 as u->0
To find the limit as \(x\) approaches \((\frac{\pi}{2})^{-}\) of \((x-\frac{\pi}{2})\sec(x)\), we need to consider the behavior of the function as \(x\) approaches \((\frac{\pi}{2})^{-}\), which means \(x\) approaches \(\frac{\pi}{2}\) from the left-hand side.
Let's break down the solution step by step:
Step 1: Substitute \(x\) with \((\frac{\pi}{2})^{-}\) in the expression:
\(( (\frac{\pi}{2})^{-} - \frac{\pi}{2} ) \sec((\frac{\pi}{2})^{-})\)
Step 2: Find the limit of the secant function as \((\frac{\pi}{2})^{-}\):
The secant function is equal to \(\frac{1}{\cos(x)}\). As \(x\) approaches \((\frac{\pi}{2})^{-}\), \(\cos(x)\) approaches \(0\) from the left side. Therefore, the limit of \(\sec(x)\) as \(x\) approaches \((\frac{\pi}{2})^{-}\) is \(+\infty\).
Step 3: Back to the original expression:
\(( (\frac{\pi}{2})^{-} - \frac{\pi}{2} ) \cdot (+\infty)\)
Step 4: Simplify the expression:
Since we have an indeterminate form of \((\frac{-\infty}{\infty})\), we need to manipulate the expression further.
Let \(a = (\frac{\pi}{2})^{-}-\frac{\pi}{2}\).
Then, the expression becomes: \((a) \cdot (+\infty)\)
Step 5: Analyze the value of \(a\):
\(a\) is a positive value since we are subtracting a positive number from \((\frac{\pi}{2})^{-}\), which is a number less than \(\frac{\pi}{2}\). Hence, \(a > 0\).
Step 6: Determine the final result:
Multiplying a positive value (\(a\)) with positive infinity, we get a result of \(+\infty\).
Therefore, the limit as \(x\) approaches \((\frac{\pi}{2})^{-}\) of \((x-\frac{\pi}{2})\sec(x)\) is \(+\infty\).
In order to find the limit as x approaches (π/2)^- of (x - π/2) * sec(x), we can follow these steps:
Step 1: Simplify the expression
Let's expand the expression:
(x - π/2) * sec(x) = x * sec(x) - (π/2) * sec(x)
Step 2: Apply the limit
Next, we'll apply the limit operator to each term separately since x is approaching (π/2)^-:
lim (x->(π/2)^-) x * sec(x) - lim (x->(π/2)^-) (π/2) * sec(x)
Step 3: Evaluate the limits
To evaluate the limits, we need to know the limits of x * sec(x) and (π/2) * sec(x) as x approaches (π/2)^-.
Let's calculate these limits one by one:
For the first term, lim (x->(π/2)^-) x * sec(x):
This is an indeterminate form, as we have x multiplying an unbounded function (sec(x)) as x approaches (π/2)^-. To resolve this, we can use L'Hôpital's rule.
Applying L'Hôpital's rule:
lim (x->(π/2)^-) x * sec(x) = lim (x->(π/2)^-) [x * (d/dx(sec(x)) / d/dx(1))]
Differentiating sec(x):
= lim (x->(π/2)^-) x * [(sec(x) * tan(x)) / 1]
Evaluating the limit now becomes:
lim (x->(π/2)^-) x * sec(x) = lim (x->(π/2)^-) [x * sec(x) * tan(x)]
For the second term, lim (x->(π/2)^-) (π/2) * sec(x):
The sec(x) function is not bounded, and as x approaches (π/2)^-, sec(x) approaches infinity. Therefore, this limit is also infinity.
Step 4: Combine the limits
Now that we have evaluated the limits of each term, we can combine the results:
lim (x->(π/2)^-) x * sec(x) - lim (x->(π/2)^-) (π/2) * sec(x) = [lim (x->(π/2)^-) x * sec(x) * tan(x)] - [lim (x->(π/2)^-) (π/2) * sec(x)]
Since the limit of the second term is infinity, and the first term is still an indeterminate form, we cannot determine the limit of the given expression as x approaches (π/2)^- using these steps.
Additional techniques, such as series expansion or more advanced mathematical tools, may be needed to evaluate the limit.