Question Statement

Find the points of discontinuity of the function:

$f(x)=\frac{2x}{x^2+5}$

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Background and Explanation

Rational functions (ratios of polynomials) are continuous everywhere except where the denominator equals zero. To find discontinuities, we must determine if there exist any real values of $x$ that make the denominator zero.

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Solution

We begin by analyzing the given function:

$f(x)=\frac{2x}{x^2+5}$

Step 1: Identify the function type

Since $f(x)$ is expressed as a ratio of two polynomials ($2x$ and $x^2+5$), it is a rational function.

Step 2: Analyze the denominator

For rational functions, discontinuities occur only where the denominator equals zero. Let's examine the denominator:

$x^2+5$

We need to determine if there exists any real number $x$ such that $x^2+5=0$.

Step 3: Check for real roots

Solving for when the denominator equals zero:

$x^2+5=0$ $x^2=-5$

Since $x^2\ge 0$ for all real numbers $x$, and $5>0$, we have: $x^2+5\ge 5>0\text{for all real }x$

Therefore, $x^2+5\ne 0$ for any real number $x$.

Step 4: Conclusion

Since the denominator never vanishes for any real value of $x$, the function $f(x)$ is defined and continuous for all real numbers.

Conclusion: $f(x)$ has no points of discontinuity (it is continuous on $\mathbb{R}$).

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Key Formulas or Methods Used

• Rational Function Continuity: A rational function $\frac{P(x)}{Q(x)}$ is continuous everywhere except where $Q(x)=0$
• Domain Analysis: Checking for real solutions to $x^2+5=0$ to identify potential discontinuities
• Properties of Squares: $x^2\ge 0$ for all $x\in \mathbb{R}$, implying $x^2+5\ge 5$

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Summary of Steps

1. Identify the function type: Recognize $f(x)$ as a rational function (ratio of polynomials)
2. Locate potential discontinuities: Set the denominator equal to zero: $x^2+5=0$
3. Solve for critical values: Rearrange to get $x^2=-5$, which has no real solutions since squares of real numbers are non-negative
4. Conclude continuity: Since no real $x$ makes the denominator zero, the function is continuous for all real numbers with no points of discontinuity