Find the limit. lim t-->∞ [(t^2 -t)/(t-1)I +√((t+8))j + sin(πt)/ln(t)k]
huh.
clearly, (t^2-t)/(t-1) = t and goes to ∞
likewise, √(t+8) -> ∞
since ln(t) -> ∞ and |sin(πt)| <= 1, the quotient -> 0
To find the limit as t approaches infinity of the given expression, we need to examine the behavior of each component separately.
Let's break down the expression:
L = lim t-->∞ [(t^2 - t)/(t - 1)I + √(t + 8)j + sin(πt)/ln(t)k]
The first component [(t^2 - t)/(t - 1)] represents the magnitude in the direction of the first basis vector, I. To determine its limit, we simplify the expression:
[(t^2 - t)/(t - 1)] = [(t - 1) * t / (t - 1)] = t
As t approaches infinity, t also approaches infinity. Therefore, the limit of the first component is infinity in the I direction.
The second component, √(t + 8)j, represents the magnitude in the direction of the second basis vector, j. As t approaches infinity, (t + 8) also approaches infinity. The square root of a number approaching infinity will also approach infinity. Hence, the limit of the second component is infinity in the j direction.
The third component, sin(πt)/ln(t)k, represents the magnitude in the direction of the third basis vector, k. We need to evaluate the limit of this expression as t approaches infinity:
lim t-->∞ [sin(πt)/ln(t)]
For this component, we have an indeterminate form of "0/0" since both the numerator and denominator approach zero. To resolve this, we can apply L'Hôpital's rule by differentiating the numerator and denominator with respect to t:
lim t-->∞ [π*cos(πt) / (1 / t)] = lim t-->∞ (π*cos(πt)*t)
At this point, it is important to note that the cosine function oscillates between -1 and 1. While πt continues to increase indefinitely as t approaches infinity, the oscillating cosine function will cause the expression to fluctuate between positive and negative values, without a clear limit.
As a result, the limit of the third component does not exist, and the third basis vector, k, does not contribute to the overall limit of the expression.
In summary, the limit of the given expression as t approaches infinity is infinity in the I direction and infinity in the j direction. The k component does not affect the overall limit.