Calculus Examples

$f (x) = {\begin{cases} x^{2} + 1 & x < 1 \\ 2 & x = 1 \\ 7 - 5 x & x > 1 \end{cases}$

Step 1

Find the limit of $x^{2} + 1$ as $x$ approaches $1$ from the left.

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Step 1.1

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

$lim_{x \to 1^{-}} x^{2} + 1$

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 $1$ .

$lim_{x \to 1^{-}} x^{2} + lim_{x \to 1^{-}} 1$

Step 1.2.2

Move the exponent $2$ from $x^{2}$ outside the limit using the Limits Power Rule.

${(lim_{x \to 1^{-}} x)}^{2} + lim_{x \to 1^{-}} 1$

Step 1.2.3

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

${(lim_{x \to 1^{-}} x)}^{2} + 1$

Step 1.3

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

$1^{2} + 1$

Step 1.4

Simplify the answer.

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Step 1.4.1

One to any power is one.

$1 + 1$

Step 1.4.2

Add $1$ and $1$ .

$2$

Step 2

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

$2$

Step 3

Since the limit of $x^{2} + 1$ as $x$ approaches $1$ from the left is equal to the function value at $x = 1$ , the function is continuous at $x = 1$ .

Continuous

Step 4

Find the limit of $7 - 5 x$ as $x$ approaches $1$ from the right.

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Step 4.1

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

$lim_{x \to 1^{+}} 7 - 5 x$

Step 4.2

Evaluate the limit.

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Step 4.2.1

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

$lim_{x \to 1^{+}} 7 - lim_{x \to 1^{+}} 5 x$

Step 4.2.2

Evaluate the limit of $7$ which is constant as $x$ approaches $1$ .

$7 - lim_{x \to 1^{+}} 5 x$

Step 4.2.3

Move the term $5$ outside of the limit because it is constant with respect to $x$ .

$7 - 5 lim_{x \to 1^{+}} x$

Step 4.3

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

$7 - 5 \cdot 1$

Step 4.4

Simplify the answer.

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Step 4.4.1

Multiply $- 5$ by $1$ .

$7 - 5$

Step 4.4.2

Subtract $5$ from $7$ .

$2$

Step 5

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

$2$

Step 6

Since the limit of $7 - 5 x$ as $x$ approaches $1$ from the right is equal to the function value at $x = 1$ , the function is continuous at $x = 1$ .

Continuous

Step 7