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Find the limit \lim_{t \rightarrow \infty} \frac{t - t\sqrt t}{2t^{3/2} +3t-5}

Question:

Find the limit {eq}\lim_{t \rightarrow \infty} \frac{t - t\sqrt t}{2t^{3/2} +3t-5} {/eq}

The Limit in Calculus:

The limit of a function {eq}f(t) {/eq} can be written as: {eq}\displaystyle\lim_{x\rightarrow \, a} f(t) {/eq}

To solve this problem, we'll divide the expression by the highest denominator power in order to get a simpler form, then plug in the value of {eq}t {/eq} to get the solution.

Answer and Explanation:

We are given:

{eq}\lim_{t \rightarrow \infty} \frac{t - t\sqrt t}{2t^{3/2} +3t-5} {/eq}

Divide by highest denominator:

{eq}=\lim_{t \rightarrow \infty} \dfrac{\frac{1}{t^{1/2}}- 1}{2 +\frac{3}{t^{1/2}}-\frac{5}{x^{3/2}}} {/eq}

{eq}=\dfrac{\lim_{t \rightarrow \infty} \left( \frac{1}{t^{1/2}}- 1 \right) }{\lim_{t \rightarrow \infty} \left( 2 +\frac{3}{t^{1/2}}-\frac{5}{x^{3/2}} \right)} {/eq}

{eq}=\dfrac{1}{2} {/eq}

Therefore the solution is {eq}\lim_{t \rightarrow \infty} \frac{t - t\sqrt t}{2t^{3/2} +3t-5} =\dfrac{1}{2} {/eq}


Learn more about this topic:

Understanding the Properties of Limits

from Math 104: Calculus

Chapter 5 / Lesson 5
9.2K

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