Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

1. Limits and Continuity

Finding Limits Algebraically

Calculus

1. Limits and Continuity

Finding Limits Algebraically

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1:55 minutes

Problem 80b

Textbook Question

Find the vertical asymptotes. For each vertical asymptote x=a, analyze lim x→a^− f(x) and lim x→a^+f(x).

f(x)=3e^x+10 / e^x

Verified step by step guidance

Step 1: Identify the form of the function f(x) = \frac{3e^x + 10}{e^x}. This is a rational function where the numerator is 3e^x + 10 and the denominator is e^x.

Step 2: Determine where the denominator is zero, as vertical asymptotes occur where the function is undefined. Set e^x = 0 and solve for x. However, e^x is never zero for any real number x, so there are no vertical asymptotes from the denominator being zero.

Step 3: Simplify the function if possible. Notice that f(x) = \frac{3e^x + 10}{e^x} can be rewritten as f(x) = 3 + \frac{10}{e^x}.

Step 4: Analyze the behavior of f(x) as x approaches positive and negative infinity. As x → ∞, e^x → ∞, so \frac{10}{e^x} → 0, and f(x) approaches 3. As x → -∞, e^x → 0, so \frac{10}{e^x} → ∞, and f(x) approaches ∞.

Step 5: Conclude that there are no vertical asymptotes for this function, but there is a horizontal asymptote at y = 3 as x approaches positive infinity.

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Key Concepts

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

Vertical Asymptotes

Vertical asymptotes occur in a function when the output approaches infinity as the input approaches a certain value from either the left or the right. This typically happens when the function is undefined at that point, often due to division by zero. Identifying vertical asymptotes involves finding values of x that make the denominator zero while ensuring the numerator is not also zero at those points.

Limits

Limits are fundamental in calculus, representing the value that a function approaches as the input approaches a certain point. In the context of vertical asymptotes, we analyze the left-hand limit (as x approaches a from the left) and the right-hand limit (as x approaches a from the right) to determine the behavior of the function near the asymptote. This helps in understanding whether the function tends to positive or negative infinity.

Exponential Functions

Exponential functions, such as f(x) = 3e^x + 10 / e^x, involve a constant base raised to a variable exponent. These functions exhibit rapid growth or decay, depending on the base and the exponent's sign. Understanding the behavior of exponential functions is crucial for analyzing limits and asymptotic behavior, especially since they can dominate polynomial or rational functions in terms of growth rates.

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Related Practice

Textbook Question

Find the vertical asymptotes. For each vertical asymptote x=a, analyze lim x→a^− f(x) and lim x→a^+f(x).

f(x)=√x^2+2x+6−3 / x−1

Textbook Question

Analyze lim x→∞ f(x) and lim x→−∞ f(x), and then identify any horizontal asymptotes.

f(x)=|1−x^2| / x(x+1)

Textbook Question

Find the vertical asymptotes. For each vertical asymptote x=a, analyze lim x→a^− f(x) and lim x→a^+f(x).

f(x)=|1−x^2| / x(x+1)

Textbook Question

Analyze lim x→∞ f(x) and lim x→−∞ f(x), and then identify any horizontal asymptotes.

f(x)=3e^x+10 / e^x

Textbook Question

Analyze lim x→∞ f(x) and lim x→−∞ f(x), and then identify any horizontal asymptotes.

f(x)=cos x+2√x / √x.

Textbook Question

Find the vertical asymptotes. For each vertical asymptote x=a, analyze lim x→a^− f(x) and lim x→a^+f(x).

f(x)=cos x+2√x / √x.

Textbook Question

The hyperbolic cosine function, denoted $\cosh (x)$ , is used to model the shape of a hanging cable (a telephone wire, for example). It is defined as $\cosh (x) = \frac{e^{x} + e^{- x}}{2}$ .