Determine the following limits.lim⁡x→1+x2−5x+6x−1{\displaystyle\lim_{x\$$to1^{+}$$}}\frac{$$x^2-5x+6$$}{x-1}

Ch. 2 - Limits

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 2 - Limits

Problem 2.4.35

Chapter 2, Problem 2.4.35

Determine the following limits.

$\lim_{x \to 1^{+}} \frac{x^{2} - 5 x + 6}{x - 1}$

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Identify the limit expression: $ \lim_{x \to 1^{+}} \frac{x^2 - 5x + 6}{x - 1} $.

Factor the numerator $ x^2 - 5x + 6 $. Notice that it can be factored into $(x - 2)(x - 3)$.

Rewrite the limit expression using the factored form: $ \lim_{x \to 1^{+}} \frac{(x - 2)(x - 3)}{x - 1} $.

Observe that direct substitution of $ x = 1 $ in the denominator results in division by zero, indicating a potential removable discontinuity.

Since the limit is from the right ($ x \to 1^{+} $), evaluate the behavior of the function as $ x $ approaches 1 from the positive side, simplifying the expression if possible.

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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 as x approaches 1 from the right, which is denoted as x → 1⁺.

Factoring Polynomials

Factoring polynomials involves rewriting a polynomial as a product of simpler polynomials or factors. This technique is essential for simplifying expressions, especially when evaluating limits. In the given limit, the numerator x² - 5x + 6 can be factored to identify any common factors with the denominator, which aids in resolving the limit.

L'Hôpital's Rule

L'Hôpital's Rule is a method used to evaluate limits that result in indeterminate forms, such as 0/0 or ∞/∞. It states that if the limit of f(x)/g(x) results in an indeterminate form, the limit can be found by taking the derivative of the numerator and the derivative of the denominator. This rule may be applicable if direct substitution in the limit leads to an indeterminate form.

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Textbook Question

Determine the following limits.

$\lim_{θ \to 0^{-}} \frac{\sin θ}{\cos^{2} θ - 1}$

Textbook Question

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