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Math  /  Calculus

Question[3 marks] (b) Consider a function f(x)=12−xf(x)=\frac{1}{2-\sqrt{x}}. Find lim⁡x→∞f(x)\lim _{x \rightarrow \infty} f(x) and state the equation of horizontal asymptotes for f(x)f(x). [3 marks]

Studdy Solution

STEP 1

1. We are given the function f(x)=12−x f(x) = \frac{1}{2 - \sqrt{x}} .
2. We need to find the limit of f(x) f(x) as x→∞ x \rightarrow \infty .
3. We need to determine the equation of any horizontal asymptotes for f(x) f(x) .

STEP 2

1. Analyze the behavior of x \sqrt{x} as x→∞ x \rightarrow \infty .
2. Determine the limit of the denominator 2−x 2 - \sqrt{x} as x→∞ x \rightarrow \infty .
3. Calculate the limit of f(x)=12−x f(x) = \frac{1}{2 - \sqrt{x}} as x→∞ x \rightarrow \infty .
4. State the equation of the horizontal asymptote based on the limit.

STEP 3

Analyze the behavior of x \sqrt{x} as x→∞ x \rightarrow \infty :
As x→∞ x \rightarrow \infty , x→∞ \sqrt{x} \rightarrow \infty .

STEP 4

Determine the limit of the denominator 2−x 2 - \sqrt{x} as x→∞ x \rightarrow \infty :
lim⁡x→∞(2−x)=2−∞=−∞ \lim_{x \rightarrow \infty} (2 - \sqrt{x}) = 2 - \infty = -\infty

STEP 5

Calculate the limit of f(x)=12−x f(x) = \frac{1}{2 - \sqrt{x}} as x→∞ x \rightarrow \infty :
lim⁡x→∞f(x)=lim⁡x→∞12−x=1−∞=0 \lim_{x \rightarrow \infty} f(x) = \lim_{x \rightarrow \infty} \frac{1}{2 - \sqrt{x}} = \frac{1}{-\infty} = 0

STEP 6

State the equation of the horizontal asymptote based on the limit:
Since lim⁡x→∞f(x)=0 \lim_{x \rightarrow \infty} f(x) = 0 , the horizontal asymptote is y=0 y = 0 .
The limit of f(x) f(x) as x→∞ x \rightarrow \infty is 0 0 , and the equation of the horizontal asymptote is:
y=0 y = 0

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