Question

Compute the limit of the problem in the picture

Answer 1

1/2

Explanation:

You want the limit ...

`\displaystyle \lim_(t\to\infty){(t-3)/(√(4t^2+25))}`

Solution 1

In general, for polynomial-like functions, the limit is the ratio of the highest-degree terms:

`\displaystyle \lim_(t\to\infty){(t-3)/(√(4t^2+25))} = \lim_(t\to\infty){(t)/(√(4t^2))}=(t)/(2t)=\boxed{(1)/(2)}`

Solution 2

You can consider the limit when x→0 and t=1/x.

`\displaystyle \lim_(t\to\infty){(t-3)/(√(4t^2+25))} = \lim_(x\to 0){\frac{(1)/(x)-3}{\sqrt{(4)/(x^2)+25}}}\\\\\\=\lim_(x\to 0)(1-3x)/(√(4+25x^2))=(1)/(√(4))=\boxed{(1)/(2)}`