Evaluate each limit. lim x → 3 x 2 + 7 {\displaystyle\lim_{x\to3}\sqrt{x^2+7}{}}

Welcome back, everyone. Evaluate the limit. Limit as X approaches 5 of square root of X2 + 11, and we're given 4 answer choices AS 5, B6, C7, and D 8. So let's look at the given limit. Let's rewrite it. We have limit as X approaches 5 of square root of X2 + 11. And let's recall that we begin with direct substitution. This gives us square root of 52 + 11. If this is a finite number, then we are done. We just don't want to get any indeterminate forms. Now square root of 25 plus 11 gives us square root of 36, which is simply 6. This is not an indeterminate form because it is a finite number, and we can conclude that the correct answer to this problem corresponds to the answer choice B, which is 6. Thank you for watching.

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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
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- Area Between Curves
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10. Physics Applications of Integrals 3h 16m
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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
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- Taylor Series & Taylor Polynomials
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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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Problem 2.6.53

Textbook Question

Evaluate each limit.

$\lim_{x \to 3} \sqrt{x^{2} + 7}$

Verified step by step guidance

Identify the limit expression: $ \lim_{x \to 3} \sqrt{x^2 + 7} $.

Recognize that this is a direct substitution problem since the function $ \sqrt{x^2 + 7} $ is continuous at $ x = 3 $.

Substitute $ x = 3 $ directly into the expression: $ \sqrt{3^2 + 7} $.

Simplify the expression inside the square root: $ 3^2 + 7 = 9 + 7 $.

Calculate the square root of the simplified expression: $ \sqrt{16} $.

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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 specific points, which is crucial for evaluating continuity and differentiability. In this case, we are interested in the limit of the function as x approaches 3.

Continuous Functions

A function is continuous at a point if the limit of the function as it approaches that point equals the function's value at that point. This concept is essential for evaluating limits, as it allows us to directly substitute the value into the function if it is continuous. The function in the limit question, √(x² + 7), is continuous everywhere since it is a polynomial under a square root.

Substitution in Limits

Substitution is a technique used in limit evaluation where you replace the variable in the function with the value it approaches. If the function is continuous at that point, this method yields the limit directly. For the given limit, substituting x = 3 into the function √(x² + 7) will provide the limit value without further manipulation.