Determine where [tex]$\frac{2}{\pi x+1}$[/tex] is not continuous.
Evaluate [tex]$\lim _{x \rightarrow 0} \frac{\sin 3 x}{x}$[/tex].
The vertical asymptotes of [tex]$\frac{x^3+x}{x^2-4}$[/tex] is at [tex]$x=$[/tex] ?
Find the vertical asymptotes of [tex]$f(x)=\frac{2 x^3+9}{(x-1)^2}$[/tex].
What is the vertical asymptote of [tex]$\frac{9}{x^2-9}$[/tex] ?
Evaluate [tex]$\lim _{x \rightarrow \infty} \frac{3-4 x^2}{2 x^2-x-2}$[/tex]
The following are turning points except:
Evaluate [tex]$\lim _{x \rightarrow 1^{+}} \frac{5}{x-1}$[/tex]
Answer (1)
28: Find the point of discontinuity by setting the denominator to zero: π x + 1 = 0 , which gives x = − π 1 .
29: Evaluate the limit using L'Hopital's rule: lim x → 0 x s i n 3 x = lim x → 0 1 3 c o s 3 x = 3 .
30: Find vertical asymptotes by setting the denominator to zero: x 2 − 4 = 0 , which gives x = ± 2 .
31: Find vertical asymptotes by setting the denominator to zero: ( x − 1 ) 2 = 0 , which gives x = 1 .
32: Find vertical asymptotes by setting the denominator to zero: x 2 − 9 = 0 , which gives x = ± 3 .
33: Evaluate the limit at infinity: lim x → ∞ 2 x 2 − x − 2 3 − 4 x 2 = − 2 .
34: Identify non-turning points: Non-stationary point.
35: Evaluate the limit from the right: lim x → 1 + x − 1 5 = + ∞ .
x = − π 1 , 3 , x = 2 , − 2 , x = 1 , x = − 3 , 3 , − 2 , Non-stationary point , + ∞
Explanation
Introduction We are given a series of calculus problems. Let's solve them one by one.
Question 28 Solution Question 28: Determine where π x + 1 2 is not continuous. A rational function is discontinuous where the denominator is zero. So, we set π x + 1 = 0 and solve for x : π x = − 1 x = − π 1
Question 29 Solution Question 29: Evaluate lim x → 0 x s i n 3 x . We can use L'Hopital's rule since the limit is in the indeterminate form 0 0 . Taking the derivative of the numerator and denominator, we get: x → 0 lim 1 3 cos 3 x = 3 cos ( 0 ) = 3
Question 30 Solution Question 30: The vertical asymptotes of x 2 − 4 x 3 + x is at x = ? Vertical asymptotes occur where the denominator is zero and the numerator is non-zero. We factor the denominator: x 2 − 4 = ( x − 2 ) ( x + 2 ) The denominator is zero at x = 2 and x = − 2 . The numerator is x 3 + x = x ( x 2 + 1 ) . The numerator is not zero at x = 2 or x = − 2 . Therefore, the vertical asymptotes are at x = 2 and x = − 2 .
Question 31 Solution Question 31: Find the vertical asymptotes of f ( x ) = ( x − 1 ) 2 2 x 3 + 9 . Vertical asymptotes occur where the denominator is zero. So, we set ( x − 1 ) 2 = 0 and solve for x : x − 1 = 0 x = 1
Question 32 Solution Question 32: What is the vertical asymptote of x 2 − 9 9 ? Vertical asymptotes occur where the denominator is zero. We set x 2 − 9 = 0 and solve for x : x 2 = 9 x = ± 3 So, the vertical asymptotes are at x = 3 and x = − 3 .
Question 33 Solution Question 33: Evaluate lim x → ∞ 2 x 2 − x − 2 3 − 4 x 2 . To evaluate this limit, we divide the numerator and denominator by the highest power of x , which is x 2 : x → ∞ lim 2 − x 1 − x 2 2 x 2 3 − 4 As x approaches infinity, x 2 3 , x 1 , and x 2 2 approach 0. Therefore, the limit is: 2 − 0 − 0 0 − 4 = 2 − 4 = − 2
Question 34 Solution Question 34: The following are turning points except. Turning points are maximum points, minimum points, and inflection points. A non-stationary point is not necessarily a turning point.
Question 35 Solution Question 35: Evaluate lim x → 1 + x − 1 5 . As x approaches 1 from the right, x − 1 approaches 0 from the positive side. Therefore, x − 1 5 approaches positive infinity.
Final Answers Final Answers: 28: x = − π 1 29: 3 30: x = 2 , − 2 31: x = 1 32: x = − 3 , 3 33: -2 34: Non-stationary point 35: + ∞
Examples
Understanding limits and continuity is crucial in various fields. For instance, in physics, when analyzing the motion of an object, knowing where a function representing its position is continuous helps predict its trajectory. Similarly, in economics, understanding limits can help predict the behavior of markets as certain parameters approach extreme values. These concepts provide a foundation for modeling and predicting real-world phenomena.
