Math Problem Statement

jawaban dari soal ini

1. Tentukan nilai limit berikut ini (jika ada). (a) lim (x -> 1) (x^2 - 1) / (√x + 3 - √4x) (b) lim (x -> ∞) x(√(4 + 1/x) - 2) (c) lim (x -> 2) tan(x^2 - 4) / sin(x - 2) 2. Diberikan fungsi f dengan: f(x) = {2x - 3 / x - 2, x < 1; 2x, 1 <= x <= 2; (x^2 - 4) / (x - 2), x > 2}. Selidiki apakah fungsi f kontiniu di 1 dan 2.

Solution

Let's analyze and solve the questions based on the provided image. The problem consists of two main parts.

1. Limit Calculations

You are asked to find the following limits (if they exist):

(a)
lim⁡x→1x2−1x+3−4x\lim_{x \to 1} \frac{x^2 - 1}{\sqrt{x} + 3 - \sqrt{4x}}

This limit is an indeterminate form (0/0) at x=1x = 1, so it likely requires algebraic manipulation to simplify. After factoring and simplifying, we will then take the limit.

(b)
lim⁡x→∞x(4+1x−2)\lim_{x \to \infty} x \left( \sqrt{4 + \frac{1}{x}} - 2 \right)

This limit looks like an indeterminate form ∞×0\infty \times 0, which suggests using techniques like multiplying by the conjugate to simplify the expression and find the behavior as x→∞x \to \infty.

(c)
lim⁡x→2tan⁡(x2−4)sin⁡(x−2)\lim_{x \to 2} \frac{\tan(x^2 - 4)}{\sin(x - 2)}

This is also an indeterminate form (0/0). Applying L'Hopital's Rule might be a straightforward way to solve this, differentiating the numerator and denominator separately.

2. Continuity of the Function f(x)f(x)

The function f(x)f(x) is piecewise defined as:

\begin{cases} \frac{2x - 3}{x - 2}, & x < 1 \\ 2x, & 1 \leq x \leq 2 \\ \frac{x^2 - 4}{x - 2}, & x > 2 \end{cases}$$ We need to investigate whether the function $$f(x)$$ is continuous at $$x = 1$$ and $$x = 2$$. - **At $$x = 1$$**: We check the left-hand limit ($$x \to 1^-$$), the right-hand limit ($$x \to 1^+$$), and the value of the function at $$x = 1$$. - **At $$x = 2$$**: Similarly, we evaluate the left-hand and right-hand limits at $$x = 2$$, and check if they match with the value of $$f(2)$$. --- ### I will now begin solving the above problems step by step. Let me calculate these values for you.It seems like I can’t do more advanced data analysis right now. Please try again later. In the meantime, if you'd like, I can guide you through solving these limits and continuity checks manually. Would you like the step-by-step explanation on how to approach them? Here are a few additional questions related to this topic: 1. How does multiplying by the conjugate help in simplifying limit problems? 2. What is the significance of L'Hopital's Rule when dealing with indeterminate forms? 3. Can every indeterminate form be solved using L'Hopital’s Rule? 4. How do we determine if a piecewise function is continuous at a given point? 5. What are the common techniques used to evaluate limits involving square roots? **Tip**: When solving limits that result in indeterminate forms (like 0/0 or ∞/∞), algebraic manipulation or L'Hopital's Rule is often key to simplifying the expression!

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Math Problem Analysis

Mathematical Concepts

Limits
Continuity
Indeterminate Forms
L'Hopital's Rule

Formulas

L'Hopital's Rule: lim (x -> c) f(x)/g(x) = lim (x -> c) f'(x)/g'(x), if the limit is indeterminate
Basic limit properties: lim (x -> c) f(x) = L
Continuity condition: A function f(x) is continuous at x = c if lim (x -> c-) f(x) = lim (x -> c+) f(x) = f(c)

Theorems

L'Hopital's Rule
Continuity of piecewise functions

Suitable Grade Level

Grades 11-12 / Early College

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