Ch. 2 - Limits

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 2 - Limits

Problem 2.37

Chapter 2, Problem 2.37

Evaluate each limit and justify your answer.
lim x→5 ln 6(√x^2−16−3) / 5x−25

Verified step by step guidance

Step 1: Recognize that the limit involves a rational function where both the numerator and the denominator approach 0 as x approaches 5. This is an indeterminate form of type 0/0, which suggests the use of L'Hôpital's Rule.

Step 2: Apply L'Hôpital's Rule, which states that if the limit of f(x)/g(x) as x approaches a value results in an indeterminate form 0/0 or ∞/∞, then the limit can be found by taking the derivative of the numerator and the derivative of the denominator separately.

Step 3: Differentiate the numerator ln(6(√x^2−16)−3) with respect to x. Use the chain rule and the derivative of the natural logarithm function. The derivative of ln(u) is 1/u * du/dx, where u = 6(√x^2−16)−3.

Step 4: Differentiate the denominator 5x−25 with respect to x. The derivative is straightforward since it is a linear function, resulting in 5.

Step 5: Evaluate the new limit using the derivatives obtained in Steps 3 and 4. Substitute x = 5 into the resulting expression to find the limit.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Limits

Limits are fundamental in calculus, representing the value that a function approaches as the input approaches a certain point. They are essential for understanding continuity, derivatives, and integrals. In this question, evaluating the limit as x approaches 5 involves determining the behavior of the function near that point, which may require algebraic manipulation or applying limit laws.

Natural Logarithm

The natural logarithm, denoted as ln, is the logarithm to the base e (approximately 2.718). It is a crucial function in calculus, particularly in problems involving growth rates and exponential functions. In the given limit, the natural logarithm of an expression is involved, which may affect the limit's evaluation, especially if the argument approaches zero or infinity.

Indeterminate Forms

Indeterminate forms occur in calculus when evaluating limits leads to expressions like 0/0 or ∞/∞, which do not provide clear information about the limit's value. In this case, the limit may require techniques such as L'Hôpital's Rule or algebraic simplification to resolve. Recognizing and addressing indeterminate forms is crucial for correctly evaluating the limit presented in the question.

Recommended video:

3:56

Slope-Intercept Form

Related Practice

Textbook Question

Use analytical methods and/or a graphing utility to identify the vertical asymptotes (if any) of the following functions.

f(x)=1/ √x sec x

Textbook Question

Find the following limits or state that they do not exist. Assume a, b, c, and k are fixed real numbers.

$\lim_{x \to \infty} x \cos (x)$

Textbook Question

Sketch the graph of a function with the given properties. You do not need to find a formula for the function.

p(0) = 2,lim x→0 p(x) = 0,lim x→2 p(x) does not exist, p(2)=lim x→2^+ p(x)=1

Textbook Question

Determine the following limits.

lim θ→π/2 sin^2 θ − 5 sin θ + 4 / sin^2 θ − 1

Textbook Question

Suppose $\lim_{x \to a} f (x) = L$ and $\lim_{x \to a} g (x) = M$ . Prove that $\lim_{x \to a} (f (x) - g (x)) = L - M$ .

views

Textbook Question

Use analytical methods and/or a graphing utility to identify the vertical asymptotes (if any) of the following functions.

g(θ)=tan πθ/10

Evaluate each limit and justify your answer. lim x→5 ln 6(√x^2−16−3) / 5x−25

Verified video answer for a similar problem:

Key Concepts

Limits

Natural Logarithm

Indeterminate Forms

Evaluate each limit and justify your answer.
lim x→5 ln 6(√x^2−16−3) / 5x−25