The process of evaluating limits as they approach a specific value in the field of calculus is about determining the value that a function gravitates towards when the input (or variable) nears a particular value. This is crucial in grasping how functions behave at certain points, even at points where the function might not have a defined value.
| Topic | Problem | Solution |
|---|---|---|
| None | Evaluate each limit. \[ \lim _{x \rightarrow-3^{+… | The limit as x approaches -3 from the right is asking for the value of the function as x gets very … |
| None | \( \lim _{x \rightarrow 4}(-3 x) \) | \( \lim _{x \rightarrow 4}(-3 x) = -3(4) \) |
| None | $\lim _{x \rightarrow 5} \frac{x+2}{x^{2}-5}$ | We are given the function \(f(x) = \frac{x+2}{x^{2}-5}\) and we are asked to find the limit as x ap… |
| None | Find the limit. Use I'Hospital's Rule where appro… | We are given the limit \(\lim _{x \rightarrow 0} \frac{\tan (7 x)}{\sin (6 x)}\). |
| None | (c) MUC (d) F $\cap$ B 3. The amount of revenue g… | Given the function F(x) = -4x^2 + 20x + 150, we need to find the limit of the function \(\frac{F(x)… |
| None | Find the following limit. \[ \lim _{x \rightarrow… | The given limit is \(\lim _{x \rightarrow 0^{+}}(2 x \cot (\pi-x))\). |
| None | Find the following limit \[ \lim _{x \rightarrow … | Given the limit expression \(\lim _{x \rightarrow 0^{+}}(4 x)^{x}\), we notice that as x approaches… |
| None | Use l'Hôpital's rule to find the limit. \[ \lim _… | The given expression is \(\frac{2 x^{2}}{\cos x-1}\). However, this is not in the form of 0/0 or ∞/… |
| None | $\lim _{x \rightarrow-10^{+}}(x+16) \frac{|x+10|}… | The limit is approaching from the right side of -10. The absolute value function |x+10| will be pos… |
| None | Find the limit. \[ \lim _{x \rightarrow 1} \frac{… | We are given the limit problem \(\lim _{x \rightarrow 1} \frac{\sqrt{x^{2}+15}-4}{x-1}\). This is a… |
| None | A. $\lim _{\theta \rightarrow 0} \frac{\sin \thet… | Given the limit problem: \(\lim _{\theta \rightarrow 0} \frac{\sin \theta}{\sin (6 \theta)}\) |
| None | $\lim _{x \rightarrow 0}\left(\frac{a^{x}+b^{x}+c… | Find the limit of the given expression as x approaches 0: $\lim _{x \rightarrow 0}\left(\frac{a^{x}… |
| None | $\lim _{x \rightarrow 0}\left(\frac{1}{x(x+1)}-\f… | First, we find the derivatives of the functions in the numerator and denominator: |
| None | (1) If $f$ is a continuous function and $\lim _{x… | Find the value of f(1) given that \(\lim_{x \rightarrow 1}(3+f(x))=2\) |
| None | $\lim _{x \rightarrow 0} x \sin \left(\frac{1}{x}… | The limit of a product is the product of the limits, provided that the limits exist. In this case, … |
| None | 4) \( \lim _{x \rightarrow 1} \frac{x^{4}-1}{x-1}… | \( \lim_{x \rightarrow 1} \frac{x^{4}-1}{x-1} \) = \( \lim_{x \rightarrow 1} \frac{(x^2+1)(x^2-1)}{… |
| None | \( \lim _{x \rightarrow 0} \frac{1}{x^{2}} \) | \( \lim _{x \rightarrow 0} \frac{1}{x^{2}} = \infty \) |
| None | The initial substitution of $x=a$ yields the form… | The initial substitution of \(x=a\) yields the form \(\frac{0}{0}\). Simplify the function algebrai… |
| None | Evaluate the following limits. \[ f(x)=\left\{\be… | Evaluate the limit from the left: \(\lim_{x \rightarrow 0^{-}} f(x) = -0 + 6 = 6\) |
| None | Show that the limit that follows does not exist b… | Given the function \(f = \frac{x^{2}y}{x^{4} + y^{2}}\), we are asked to show that the limit as (x,… |
| None | Provided $\lim _{x \rightarrow-1} f(x)=5$ and $\l… | Given that \(\lim _{x \rightarrow-1} f(x)=5\) and \(\lim _{x \rightarrow-1} g(x)=8\) |
| None | 함수 $f(x)=\left\{\begin{array}{cc}-x+3 & (x \geq 3… | 함수 $f(x)=\left\{\begin{array}{cc}-x+3 & (x \geq 3) \\ x-3 & (x<3)\end{array}\right.$ 에 대하여 $\lim _{… |
| None | \( \lim _{x \rightarrow 0} \frac{\sin 2 x}{5 x} \) | \( \lim _{x \rightarrow 0} \frac{\sin 2 x}{5 x} = \frac{1}{5} \lim _{x \rightarrow 0} \frac{\sin 2 … |
| None | $\lim _{x \rightarrow-1} \frac{x^{3}+1}{x^{2}-1}$ | Podemos realizar a divisão longa. Também podemos escrever |
| None | Use l'Hôpital's rule to find the following limit.… | We are given the limit \(\lim _{x \rightarrow 1^{+}}\left(\frac{1}{\ln x}-\frac{1}{x-1}\right)\). |
| None | $\lim _{x \rightarrow 0} \frac{(\tan x-\sin x)(\t… | Use the identity \((a - b)(a + b) = a^2 - b^2\) to simplify the expression: \(\frac{( an x - sin x)… |
| None | Use the Theorem on Limits of Rational Functions t… | We are given the function \(f(x) = \frac{x^{2}-1}{8-x}\) and we are asked to find the limit as x ap… |
| None | Use l'Hôpital's Rule to find the following limit.… | First, we need to check if the limit is in the form of 0/0 or ∞/∞ to apply l'Hôpital's Rule. As x a… |
| None | $\lim _{x \rightarrow 0} \frac{4 \sin x \sin ^{2}… | Find the derivatives of the numerator and the denominator: \(\frac{d}{dx}(4 \sin x \sin^2(\frac{x}{… |
| None | 3. $[-/ 1$ Points $]$ DETAILS SCALC9 1.6.013. MY … | The function is not defined at t=6, but we can simplify the function to find the limit as t approac… |
| None | Use l'Hôpital's Rule to evaluate $\lim _{x \right… | Given the limit \(\lim _{x \rightarrow 0} \frac{7-7 \cos x}{3 x^{2}}\), we can see that it is of th… |