Calculus Examples

$f (x) = {\begin{cases} 3 x^{2}, x \geq 1 \\ A x - 4, x < 1 \end{cases}$

Step 1

Find the limit of $A x - 4$ as $x$ approaches $1$ from the left.

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Step 1.1

Change the two-sided limit into a left sided limit.

$lim_{x \to 1^{-}} A x - 4$

Step 1.2

Evaluate the limit.

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Step 1.2.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $1$ .

$lim_{x \to 1^{-}} A x - lim_{x \to 1^{-}} 4$

Step 1.2.2

Move the term $A$ outside of the limit because it is constant with respect to $x$ .

$A lim_{x \to 1^{-}} x - lim_{x \to 1^{-}} 4$

Step 1.2.3

Evaluate the limit of $4$ which is constant as $x$ approaches $1$ .

$A lim_{x \to 1^{-}} x - 1 \cdot 4$

Step 1.3

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

$A \cdot 1 - 1 \cdot 4$

Step 1.4

Simplify each term.

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Step 1.4.1

Multiply $A$ by $1$ .

$A - 1 \cdot 4$

Step 1.4.2

Multiply $- 1$ by $4$ .

$A - 4$

Step 2

Evaluate $3 x^{2}$ at $1$ .

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Step 2.1

Replace the variable $x$ with $1$ in the expression.

$3 {(1)}^{2}$

Step 2.2

Evaluate.

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Step 2.2.1

One to any power is one.

$3 \cdot 1$

Step 2.2.2

Multiply $3$ by $1$ .

$3$

Step 3

Since the limit of $A x - 4$ as $x$ approaches $1$ from the left is not equal to the function value at $x = 1$ , the function is not continuous at $x = 1$ .

Not continuous

Step 4