Question
[-11 Points] DETAILS MY NOTES SCALCET9 \( 2 . X P .6 .021 \). Find the limit. (If the limit is infinite, enter ' \( \alpha \) ' or ' \( -\infty \) ', as appropriate. If the limit does not otherwise exist, enter DN \[ \lim _{x \rightarrow \infty} \arctan \left(e^{x}\right) \]
Ask by Fitzgerald Hall. in the United States
Feb 26,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The limit is π/2.
Solution
To solve
limₓ→∞ arctan(eˣ),
we first note that as x → ∞, eˣ becomes unbounded (eˣ → ∞). The arctan function has a horizontal asymptote, namely
limₜ→∞ arctan(t) = π/2.
Thus, if we substitute t = eˣ, it follows that
limₓ→∞ arctan(eˣ) = π/2.
The final answer is π/2.
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Simplify this solution Mind Expander
When evaluating the limit of \(\arctan(e^x)\) as \(x\) approaches infinity, we first note that as \(x\) increases, \(e^x\) grows towards infinity. The arctangent function, \(\arctan(y)\), approaches \(\frac{\pi}{2}\) as \(y\) approaches infinity. Therefore, \[ \lim_{x \rightarrow \infty} \arctan(e^x) = \frac{\pi}{2}. \] So the limit is \(\frac{\pi}{2}\). Additionally, the arctangent function is continuous, meaning that we don't have any surprises or undefined behavior as \(x\) grows larger. This straightforward behavior can create a reassuring pattern in calculus problems, where limits often yield tangible results without unexpected jumps or holes!
