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Ch. 2 - Limits
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 2 - LimitsProblem 2.33
Chapter 2, Problem 2.33

Determine the following limits.
lim x→0^− 2 / tan x

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Identify the limit expression: \( \lim_{{x \to 0^-}} \frac{2}{\tan x} \).
Recognize that as \( x \to 0^- \), \( \tan x \to 0^- \) because the tangent function approaches zero from the negative side.
Understand that \( \frac{2}{\tan x} \) will approach infinity or negative infinity depending on the sign of \( \tan x \) as \( x \to 0^- \).
Since \( \tan x \to 0^- \), the expression \( \frac{2}{\tan x} \) will approach negative infinity.
Conclude that the limit is \( -\infty \) as \( x \to 0^- \).

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Key Concepts

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

Limits

A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. It helps in understanding how functions behave near points of interest, including points where they may not be defined. In this case, we are interested in the limit of the function as x approaches 0 from the left (denoted as x→0^−).
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One-Sided Limits

Tangent Function

The tangent function, tan(x), is a periodic function defined as the ratio of the sine and cosine functions: tan(x) = sin(x)/cos(x). It has specific properties, including vertical asymptotes where cos(x) = 0, which occur at odd multiples of π/2. Understanding the behavior of tan(x) near x = 0 is crucial for evaluating the limit in the question.
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Slopes of Tangent Lines

One-Sided Limits

One-sided limits refer to the value that a function approaches as the input approaches a specific point from one side only, either from the left (denoted as x→c^−) or from the right (denoted as x→c^+). In this problem, we are specifically looking at the left-hand limit as x approaches 0, which can yield different results than the right-hand limit, especially for functions with discontinuities.
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Related Practice
Textbook Question

Determine the following limits.


limθ→0−sinθcos2θ−1{\(\displaystyle\[\lim\)_{\(\theta\]\to\)0^{-}}\(\frac{\sin\theta}{\cos^2\theta-1}\)}

Textbook Question

Determine limx→∞f(x)\(\lim\)_{x\(\rightarrow\]\infty\)}f\(\left\)(x\(\right\)) and limx→−∞f(x)\(\lim\)_{x\(\rightarrow\)-\(\infty\)}f\(\left\)(x\(\right\)) for the following functions. Then give the horizontal asymptotes of ff (if any).


f(x)=x6+834x2+3x4+1f\(\left\)(x\(\right\))=\(\frac{\sqrt[3]{x^6+8}\)}{4x^2+\(\sqrt{3x^4+1}\)}

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Textbook Question

Use the precise definition of infinite limits to prove the following limits.


limx→0(1x2+1)=∞{\(\displaystyle\]\lim\)_{x\(\to\)0}}\(\left\)(\(\frac{1}{x^2}\)+1\(\right\))=\(\infty\)

Textbook Question

Explain the meaning of lim x→a f(x) =∞.

Textbook Question

Determine the following limits.


limx→1+x2−5x+6x−1{\(\displaystyle\]\lim\)_{x\(\to\)1^{+}}}\(\frac{x^2-5x+6}{x-1}\)

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Textbook Question

Evaluate lim x→1 (x^3+3x^2−3x+1).

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