Question

Evaluating limits Evaluate the following limits, where \(c\) and \(k\) are constants. $$\lim _{h \rightarrow 0} \frac{100}{(10 h-1)^{11}+2}$$

Short answer

Answer: The value of the limit is 100.

Step-by-step solution

Identify the form of the given function

The given function is of the form: $$f(h) = \frac{100}{(10h-1)^{11}+2}$$ It is a rational function, where both the numerator and the denominator are polynomials. The numerator is a constant, and the denominator is a polynomial of an odd power.

Check the behavior of the denominator when h approaches 0

We first want to investigate the behavior of the denominator as \(h\) approaches 0. To do so, we will apply the limit rules to the polynomial part: $$\lim_{h\rightarrow 0}(10h-1)^{11}$$ As we look at the expression \((10h-1)^{11}\), we can see that when \(h\) is very close to 0 (but not exactly 0), the expression inside the parenthesis will be slightly negative, and as odd powers do not change the sign of negative numbers, it will remain negative. Therefore, when \(h\) is close to 0, the expression \((10h-1)^{11}\) will be close to -1.

Plug in the behavior of the denominator into the limit

Now that we know the behavior of the denominator, we can now plug it into the limit: $$\lim_{h\rightarrow 0}\frac{100}{(10h-1)^{11}+2}=\frac{100}{-1 + 2}$$

Evaluate the limit

With the given expression, we can now evaluate the limit as \(h\) approaches \(0\): $$\frac{100}{-1 + 2}=\frac{100}{1}=100$$ Therefore, the limit is: $$\lim_{h\rightarrow 0}\frac{100}{(10 h-1)^{11}+2}=100$$

Key concepts

Rational Functions

Rational functions are mathematical expressions that involve the ratio of two polynomials. In simpler terms, they're like fractions, but instead of having regular numbers in the numerator and denominator, we have polynomials. The general form is \( \frac{P(x)}{Q(x)} \), where both \(P(x)\) and \(Q(x)\) are polynomials. These functions are important in calculus, especially when it comes to evaluating limits, as they often exhibit interesting behavior near points where the denominator is zero.

For instance, in the exercise, we have the rational function \( \frac{100}{(10h-1)^{11}+2} \), where the numerator is a constant (100) and the denominator is a polynomial \((10h-1)^{11} + 2\). Understanding how this denominator behaves is key to solving the limit problems effectively.

Polynomial Functions

Polynomial functions are versatile mathematical expressions formed by the sum of terms, each consisting of a variable raised to a whole number power and multiplied by a coefficient. They can range from simple linear functions like \( f(x) = 2x + 3 \) to more complex ones like \( g(x) = x^4 - 3x^3 + 2x^2 + x + 1 \).

The function given in the limit problem, \((10h-1)^{11}\), is a single-term polynomial raised to an odd power. This polynomial plays a crucial role in the behavior of the rational function as \(h\) approaches 0, affecting the sign and limit value of the expression.

• When \(h\) is slightly negative, \((10h-1)\) becomes negative, and raising it to an odd power \(11\) keeps it negative.
• When \(h\) is positive but small, the expression nears zero and remains negative.

Understanding this pattern helps us simplify the limit calculation.

Limit Evaluation

Evaluating a limit involves determining what value a function approaches as the input gets closer to a certain number. It's a fundamental concept in calculus used to understand the behavior of functions at specific points, particularly where there might be discontinuities or indeterminate forms.

In this exercise, we evaluated \( \lim _{h \rightarrow 0} \frac{100}{(10h-1)^{11}+2} \). By knowing how the denominator behaves as \(h\) approaches zero, we simplified the calculation.

• First, examine how each part of the function behaves when \(h\) is near the limit point.
• Identify any simplifications or behaviors, like how expanding the denominator \((10h-1)^{11}\) remains negative, which influences the outcome.

Applying these methods correctly will lead us to the correct limit value of 100.

Odd Powers

Odd powers in polynomials refer to those where the exponent of a term is an odd number, such as 1, 3, 5, 7, and so forth. These powers have particular properties that make them fascinating in calculus. When you raise a negative number to an odd power, the result is still negative, preserving the sign of the base.

This property affected the given example significantly, since \((10h-1)^{11}\) is an odd power. Considering \(h\) approaches 0, the base \((10h-1)\) typically equals a slightly negative number close to -1, and raising it to the 11th power results in a negative number near -1 again.

• Odd powers are critical when assessing function behavior around certain limits.
• They retain the sign of the original base, which can simplify limit calculations.

Hence, understanding the nature of odd powers is essential for working with polynomial functions during limit evaluation.