Limits and Continuity Calculator

Ask and get solution to your homeworkAsk now and get step-by-step solutions

Example

Created on 2024-06-20Asked by Ava Sanchez (Solvelet student)

Determine if the function

f(x) = \begin{cases} x^2 - 4 & \text{if } x < 2 \\ 4 & \text{if } x = 2 \\ x + 2 & \text{if } x > 2 \end{cases}

is continuous at

x = 2

Solution

To determine if the function

f(x)

is continuous at

x = 2

, we check the following conditions: 1.

f(2)

is defined. 2. The limit of

f(x)

x \to 2

exists. 3. The limit of

f(x)

x \to 2

is equal to

f(2)

. 1.

f(2) = 4

is defined. 2. We find the limit from the left and right:

\lim_{x \to 2^-} f(x) = \lim_{x \to 2^-} (x^2 - 4) = 2^2 - 4 = 0

\lim_{x \to 2^+} f(x) = \lim_{x \to 2^+} (x + 2) = 2 + 2 = 4

3. Since

\lim_{x \to 2^-} f(x) \neq \lim_{x \to 2^+} f(x)

, the limit

\lim_{x \to 2} f(x)

does not exist. Therefore, the function

f(x)

is not continuous at

x = 2

. Solved on Solvelet with Basic AI Model

Some of the related questions asked by Mia Torres on Solvelet

1. Determine the value of

k

for which the function

f(x) = \begin{cases} kx, & x < 1 \\ x^2 - 1, & x \ge 1 \end{cases}

is continuous,2. Determine

\lim_{x \to \frac{\pi}{4}} \tan(x)

by direct substitution.,

DefinitionLimits and Continuity Calculator Is a calculator that will let you find whether the function is continuous at any particular point by using limits and the limit definition of continuity. For example, to check continuity at \( x = 1 \) for the function \( f(x) = \frac{x^2 - 1}{x - 1} \) \( \lim_{{x \to 1}} \frac{x^2 - 1}{x - 1} = \lim_{{x \to 1}} (x+1) = 2 \) ( f(x)= f(1) ) x(1) Here, ( f(1)=1+1) = 2 Therefore, ( f(1)+1 )= 4 Since ( f(1)= 2... thus, its incorrect ) Thus, ( f(x)= x2) x(1) = 1+1 = 2 And consequently f(x) is not continuous at x = 1

Related Calculators

Interpolation and Extrapolation Calculator Synthetic division of polynomials Calculator Numerical coefficients Calculator Vector Valued Functions Calculator

Need topic explanation ? Get video explanation