# What is #lim_(x->oo) x - sqrt(x^2 + 4x + 3)#?

graph{x - sqrt(x^2 + 4x + 3) [-12.73, 19.3, -11.61, 4.41]}

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The limit is

This kind of problem is usually introduced after students have worked with limits at infinity of ratios involving similar expressions. The trick here is to turn this expression into a ratio whose limit we can evaluate.

We'll use the (by now familiar) technique of rationalizing the numerator.

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To find the limit ( \lim_{x \to \infty} (x - \sqrt{x^2 + 4x + 3}) ), we can simplify the expression inside the square root:

[ \lim_{x \to \infty} (x - \sqrt{x^2 + 4x + 3}) ]

[ = \lim_{x \to \infty} \left(x - \sqrt{(x + 1)(x + 3)}\right) ]

[ = \lim_{x \to \infty} \left(x - \sqrt{x^2 + 4x + 3}\right) ]

[ = \lim_{x \to \infty} \left(x - \sqrt{x^2}\sqrt{1 + \frac{4}{x} + \frac{3}{x^2}}\right) ]

[ = \lim_{x \to \infty} \left(x - x\sqrt{1 + \frac{4}{x} + \frac{3}{x^2}}\right) ]

[ = \lim_{x \to \infty} \left(x - x\sqrt{1 + 0 + 0}\right) ]

[ = \lim_{x \to \infty} \left(x - x\sqrt{1}\right) ]

[ = \lim_{x \to \infty} \left(x - x\right) ]

[ = \lim_{x \to \infty} 0 ]

[ = 0 ]

Therefore, ( \lim_{x \to \infty} (x - \sqrt{x^2 + 4x + 3}) = 0 ).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- How do you find the limit of #1-2^-(1/x)/(1+2^-(1/x))# as x approaches #0^-#?
- How do you evaluate the limit #x^3/(tan^3(2x)# as x approaches #0#?
- Which function has a point of discontinuity at x=3? A) x-3/2x^2 -2x -12 B) x+3/x^2 -6x +9. Please Explain why you chose the answer.
- How do you prove that the function #f(x) = (x + 2x^3)^4# is continuous at x=-1?
- How do you find the limit of #[1/ln(x)] - [1/x-1]# as x approaches 1?

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