Evaluate the limits in Exercise 7 and 8.
lim (x → ∞) ∫₋ˣ^ˣ sin t dt

Verified step by step guidance

1. Recognize that the problem asks for the limit as \(x\) approaches infinity of the integral \(\int_{-x}^{x} \sin t \, dt\).
2. Recall the Fundamental Theorem of Calculus, which tells us how to evaluate definite integrals using antiderivatives.
3. Find the antiderivative of \(\sin t\), which is \(-\cos t\).
4. Evaluate the definite integral \(\int_{-x}^{x} \sin t \, dt\) by computing \([-\cos t]_{-x}^{x} = -\cos x - (-\cos(-x))\).
5. Use the fact that cosine is an even function, so \(\cos(-x) = \cos x\), and simplify the expression before taking the limit as \(x \to \infty\).

Key Concepts

Definite Integral and its Geometric Interpretation

A definite integral represents the net area under a curve between two limits. In this problem, the integral of sin(t) from -x to x measures the accumulated area under the sine curve over a symmetric interval, which is essential for understanding how the integral behaves as x approaches infinity.

Limit of a Function as x Approaches Infinity

The limit as x approaches infinity examines the behavior of a function or expression when x grows without bound. Here, evaluating lim (x → ∞) ∫₋ˣ^ˣ sin t dt requires understanding how the integral changes as the interval expands indefinitely.

Properties of the Sine Function and its Integral

The sine function is periodic and oscillates between -1 and 1. Its integral over symmetric intervals often cancels out due to its odd symmetry, which is crucial for evaluating the limit of the integral from -x to x as x tends to infinity.