The limits for this problem are given as follows:
5. -9/4.
6. 3/14.
7. 3/2.
<h3>What is a limit?</h3>
A limit is given by the value of function f(x) as x tends to a value.
<h3>What is the limit for item 5?</h3>
x goes to infinity, hence we just consider the terms of highest degree in the numerator and the denominator, and apply the limit of the constant(equals the constant), as follows:
![\lim_{t \rightarrow \infty} \frac{\sqrt{t} + 9t^2}{3t - 4t^2} = \lim_{t \rightarrow \infty} \frac{9t^2}{-4t^2} = \lim_{t \rightarrow \infty} -\frac{9}{4} = -\frac{9}{4}](https://tex.z-dn.net/?f=%5Clim_%7Bt%20%5Crightarrow%20%5Cinfty%7D%20%5Cfrac%7B%5Csqrt%7Bt%7D%20%2B%209t%5E2%7D%7B3t%20-%204t%5E2%7D%20%3D%20%5Clim_%7Bt%20%5Crightarrow%20%5Cinfty%7D%20%5Cfrac%7B9t%5E2%7D%7B-4t%5E2%7D%20%3D%20%5Clim_%7Bt%20%5Crightarrow%20%5Cinfty%7D%20-%5Cfrac%7B9%7D%7B4%7D%20%3D%20-%5Cfrac%7B9%7D%7B4%7D)
<h3>What is the limit for item 6?</h3>
Before applying the same property we did for the above item, we have to rationalize the expression, applying the subtraction of perfect squares, as follows:
![(\sqrt{49x^2 + 3x} - 7x) \times \frac{(\sqrt{49x^2 + 3x} + 7x)}{(\sqrt{49x^2 + 3x} + 7x)} = \frac{49x^2 + 3x - 49x^2}{(\sqrt{49x^2 + 3x} + 7x)} = \frac{3x}{(\sqrt{49x^2 + 3x} + 7x)}](https://tex.z-dn.net/?f=%28%5Csqrt%7B49x%5E2%20%2B%203x%7D%20-%207x%29%20%5Ctimes%20%5Cfrac%7B%28%5Csqrt%7B49x%5E2%20%2B%203x%7D%20%2B%207x%29%7D%7B%28%5Csqrt%7B49x%5E2%20%2B%203x%7D%20%2B%207x%29%7D%20%3D%20%5Cfrac%7B49x%5E2%20%2B%203x%20-%2049x%5E2%7D%7B%28%5Csqrt%7B49x%5E2%20%2B%203x%7D%20%2B%207x%29%7D%20%3D%20%5Cfrac%7B3x%7D%7B%28%5Csqrt%7B49x%5E2%20%2B%203x%7D%20%2B%207x%29%7D)
Hence the limit is given as follows:
![\lim_{x \rightarrow \infty} f(x) = \lim_{x \rightarrow \infty} \frac{3x}{(\sqrt{49x^2 + 3x} + 7x)}](https://tex.z-dn.net/?f=%5Clim_%7Bx%20%5Crightarrow%20%5Cinfty%7D%20f%28x%29%20%3D%20%5Clim_%7Bx%20%5Crightarrow%20%5Cinfty%7D%20%5Cfrac%7B3x%7D%7B%28%5Csqrt%7B49x%5E2%20%2B%203x%7D%20%2B%207x%29%7D)
At the square root in the denominator, as the x goes to infinity, we consider only the highest exponent, hence:
![\lim_{x \rightarrow \infty} f(x) = \lim_{x \rightarrow \infty} \frac{3x}{\sqrt{49x^2} + 7x} = \lim_{x \rightarrow \infty} \frac{3x}{14} = \lim_{x \rightarrow \infty} \frac{3}{14} = \frac{3}{14}](https://tex.z-dn.net/?f=%5Clim_%7Bx%20%5Crightarrow%20%5Cinfty%7D%20f%28x%29%20%3D%20%5Clim_%7Bx%20%5Crightarrow%20%5Cinfty%7D%20%5Cfrac%7B3x%7D%7B%5Csqrt%7B49x%5E2%7D%20%2B%207x%7D%20%3D%20%5Clim_%7Bx%20%5Crightarrow%20%5Cinfty%7D%20%5Cfrac%7B3x%7D%7B14%7D%20%3D%20%5Clim_%7Bx%20%5Crightarrow%20%5Cinfty%7D%20%5Cfrac%7B3%7D%7B14%7D%20%3D%20%5Cfrac%7B3%7D%7B14%7D)
<h3>What is the limit to item 7?</h3>
We also rationalize applying the subtraction of perfect squares, as follows:
![(\sqrt{x^2 + 13x} - \sqrt{x^2 + 10x}) \times \frac{(\sqrt{x^2 + 13x} + \sqrt{x^2 + 10x})}{(\sqrt{x^2 + 13x} + \sqrt{x^2 + 10x})} = \frac{x^2 + 13x - x^2 - 10x}{(\sqrt{x^2 + 13x} + \sqrt{x^2 + 10x})} = \frac{3x}{(\sqrt{x^2 + 13x} + \sqrt{x^2 + 10x})}](https://tex.z-dn.net/?f=%28%5Csqrt%7Bx%5E2%20%2B%2013x%7D%20-%20%5Csqrt%7Bx%5E2%20%2B%2010x%7D%29%20%5Ctimes%20%5Cfrac%7B%28%5Csqrt%7Bx%5E2%20%2B%2013x%7D%20%2B%20%5Csqrt%7Bx%5E2%20%2B%2010x%7D%29%7D%7B%28%5Csqrt%7Bx%5E2%20%2B%2013x%7D%20%2B%20%5Csqrt%7Bx%5E2%20%2B%2010x%7D%29%7D%20%3D%20%5Cfrac%7Bx%5E2%20%2B%2013x%20-%20x%5E2%20-%2010x%7D%7B%28%5Csqrt%7Bx%5E2%20%2B%2013x%7D%20%2B%20%5Csqrt%7Bx%5E2%20%2B%2010x%7D%29%7D%20%3D%20%5Cfrac%7B3x%7D%7B%28%5Csqrt%7Bx%5E2%20%2B%2013x%7D%20%2B%20%5Csqrt%7Bx%5E2%20%2B%2010x%7D%29%7D)
Following the same logic as the previous item, considering only the squared term as the root in the denominator.
![\lim_{x \rightarrow \infty} f(x) = \lim_{x \rightarrow \infty} \frac{3x}{(\sqrt{x^2 + 13x} + \sqrt{x^2 + 10x})} = \lim_{x \rightarrow \infty} \frac{3x}{x + x} = \lim_{x \rightarrow \infty} \frac{3x}{2x} = \lim_{x \rightarrow \infty} \frac{3}{2} = \frac{3}{2}](https://tex.z-dn.net/?f=%5Clim_%7Bx%20%5Crightarrow%20%5Cinfty%7D%20f%28x%29%20%3D%20%5Clim_%7Bx%20%5Crightarrow%20%5Cinfty%7D%20%5Cfrac%7B3x%7D%7B%28%5Csqrt%7Bx%5E2%20%2B%2013x%7D%20%2B%20%5Csqrt%7Bx%5E2%20%2B%2010x%7D%29%7D%20%3D%20%5Clim_%7Bx%20%5Crightarrow%20%5Cinfty%7D%20%5Cfrac%7B3x%7D%7Bx%20%2B%20x%7D%20%3D%20%5Clim_%7Bx%20%5Crightarrow%20%5Cinfty%7D%20%5Cfrac%7B3x%7D%7B2x%7D%20%3D%20%5Clim_%7Bx%20%5Crightarrow%20%5Cinfty%7D%20%5Cfrac%7B3%7D%7B2%7D%20%3D%20%5Cfrac%7B3%7D%7B2%7D)
More can be learned about limits at brainly.com/question/26270080
#SPJ1